Personally. I think you are disingenuous and not arguing in good faith- especially, how you wrote this beautiful background and then way you worded the last line.
It probably went more like this
Player: Hey, DM, do wood elves exist in your world? Can I play one? Have you added any restrictions as to what class/background they can be?
DM: Yes, you can play a wood elf, and they can have any class and background.
Player: So, all 11 classes are available for me to play? All their paths? All backgrounds?
DM: Yes, yes and yes. You can even tweak your background in any way you want within the parameters set in the background section of the PHB. You can even create your own from scratch, but I'll need to approve it if you do.
Player: Do you allow multiclassing and feats?
DM: Yes to both.
Player: Okay, my PC is a female wood elf rogue 1/monk 2 (you said we'd be starting at level 3). She has the 'spy' variant of the 'criminal' background. Is that all okay?
DM: Solid.
Then, without questioning the DM about the setting and cultures, you went off and wrote your elaborate backstory. Proudly, you handed him what you thought was a brilliant masterpiece that any DM should accept. You presented it to the DM and he said, "This is a nice background, but it doesn't fit the campaign and cultures". Since you stated in a prior post that DMs have to accept your background, because you used races, classes, and other options available in the campaign and, therefore, you have the right to determine the fluff, you probably got upset and called him a jerk.
See,it can be played both ways and is not a good place to start an argument (as in a discussion).
I think I owe you an apology here: that conversation didn't take place. The point I was trying to make is that it would be absurd for a DM to disallow a PC when:-
a.) every single rule set by the game and the DM has been followed to the letter, and
b.) the 'reason' it is disallowed is the absurd idea that the characters in the game know about the game mechanics.
This conversation (up to the absurd last line)
could have taken place, as each line is a true thing about this PC's creation, but it didn't play out as a single conversation in reality.
Now, that aside, let us assume that everything you stated was above board except the last line which seems intentionally designed to be prejudicial. You asked the DM about races, classes, feats, classes and multi-classing. That is common. It is also a point at which many players go off and make a character. Neither the player nor the DM brought up the limits on the setting or the limits on the player. That would make them both wrong and to blame ( I personally, avoid this. The player gets a 1-2 page handout with the overviews of the cultures including classes,variants, and subclasses that are appropriate. Then, there are similar handouts for each culture that go into a little more indepth (the player can read it or we can talk about it covering the points). Once the player finds a culture that they like, we can start talking about their character concept and background and work from there to make something that we can agree on (hopefully)). However, the DM told you to make a background without real guidance.
Sent off by the DM, you built a multi-page backstory in which "You intertwined published lore, adventure hooks that don't contradict published lore, adventure hooks specific to this adventure" and stuff you made up that didn't contradict published lore. You never confirmed limits for inventing new stuff and the DM didn't give you any guidelines (Personally, if I were the DM, there would again be plenty of discussion going on, but the DM gave you free rein). The DM gave free rein and liked your backstory. That is cool.
In reality, this was for the organised play of Princes of the Apocalypse. This campaign is set in the Forgotten Realms, a setting I've been familiar with for decades. Since 1st edition. The DM didn't have to give me a two-page summary of 'his' world; I've read literally thousands of pages about this 'campaign world'.
The specifics of the campaign world in this published campaign, as far as character creation are concerned, are available to me. I read what it said about the Elves of the High Forest faction; taking that as my faction needed no more DM approval than taking any of the other factions available to players of this campaign. Although a DM in a home game could arbitrarily limit our faction choices, he can't do that in organised play.
Oh, and I didn't actually write down my backstory; the DM and I spoke about it at length. I could easily write it down if I wanted, though.
Now, we get to the part of the monastery- In my opinion, you both are at fault for not communicating earlier. The DM is at fault for sending you off without more discussion on acceptable limits or any other guideline. You were at fault for creating an organization (and possibly a specific culture of wood elves) for the campaign without talking to the DM). Personally, I put much more of the blame on the DM for giving free rein and sending you off to make a character without any real guidelines and pretty much a blank state. However, it is still your responsibility, in my opinion, to communicate with the DM if you are going to do this (Kids this is why it is important for the player and DM to communicate *throughout* the character generation process. It keeps players and DMs on the same page!)
Yeah, I made up the Lachrymae Shevarash. The DM could've nixed that if he wanted to. But why would he? According to the rules for the Criminal background, they get a Criminal Contact. Since Spy is an official variant of Criminal, it seems that Criminal Contact would actually be Spy Contact, but as a game feature I don't expect one to be more 'powerful' an advantage than the other. In fact, it gives the DM power over me, and I'm okay about that because I chose to give him that form of power.
But who has the authority over background? In one way, the DM has the authority over everything in his game, but if he extends this to the PCs then he might as well be playing Magic Story Time, hand out pre-gens and make all the decisions on behalf of the player so that everything stays under his control.
When a player makes up a backstory, guess what? He makes stuff up!
Player: Hey DM, my backstory is that, when I was around 12 years old, my father...
DM: Father? What are you talking about? I didn't give you permission to alter MY world by adding NPCs that YOU made up! What about the delicate balance of my world? And what have you written on your character sheet? 'Right handed'? In MY world, ALL elves are left handed! Wait,
what colour hair...!
(No, this isn't a real conversation, just pointing out an absurd extreme to illustrate that just because it's the DM's world doesn't mean the players have no part in the creation of their own PC)
Of course, if the player says that his PC is actually King of the Elves, well, shoot it down. Fire at will!
So, to what extent should players be allowed to invent? It's a judgement call, but the player should not invent something that messes with the established way things work in the world, without getting the DM on board. With my elven spy, nothing about the Lachrymae Shevarash impacts the campaign unless the DM wants it to. He can just treat it as any other Elves of the High Forest faction; it's just flavour. But isn't flavour an essential part of elevating what could be essentially a really complex board game into a Role-Playing Game?
Now as for your prejudicial last line. I doubt that happens. Either you overstepped your bounds and created a new organization and the DM does not approve and says so (again, he or she is at least partially to blame) or your background is so wonderful that they a) allow your monastery; or b) the DM says to include some monastery incorporating the natural elements. Either way, it is the DM's call even if I feel the DM is at least partially or shoulders the majority of the blame (a setting/campaign issue has priority as players are told to talk to their DM). You as a player then have the choice to a) make the adjustment, b) choose to play something different, or c) walk. However, if you insist that you put all that work and are playing the character without making the change, then yes the DM has the right to say, "Sorry, you are no longer welcome" (personally, I think they are wrong because they bear a large part of the responsibility by not communicating, but it is his or her campaign and final decision).
And now to the relevance of the infamous 'last line' to this thread: there are lots of good reasons for a DM to intervene in the character creation process, but 'monks only come from monasteries' is, in context, absurd.
First, on its face, the building isn't the person. If a monastery-trained monk learned that his monastery had been destroyed, would he lose his class abilities?
Next, in game, a 'monk' can be a person trained in a monastery, but that relates to the Acolyte background, not a limit on the game mechanics of characters raised there. They could be a Commoner, Cleric, Fighter, Paladin, even *gasp*
monk in terms of game mechanics, but creatures in the game can have no concept of game mechanics.
Whether or not a PC with levels in the Monk class actually identifies as a 'monk' in game is up to the player, not the DM! The DM does not exercise mind control! He is not the Thought Police! What the player thinks is up to him.
The fluff in the class descriptions are not rules, they are mere examples to get your imaginations flowing. Stereotypes can be useful, but there is no rule limiting players to one of the three stereotypes in the descriptions for each class.
In game, there may well be an organisation that calls itself 'Shadow Monks', but the creatures in the game cannot know the game mechanics of other creatures. It's possible that non-monks (in terms of class levels) are Shadow Monks in game, by using other means that resemble those abilities. And just because one group calls itself Shadow Monks in game does not limit anyone else from taking shadow monk class levels and fluffing a backstory that doesn't involve the in game Shadow Monks in any way.