I'm going to cut out all the objectionable parts and try to respond to just the core of your questions...
So, since this "Francis the Guard" example....
Is the player correct about having been raised in an orphanage?
That depends. Was it established before play began that the player character was raised in an orphanage, or is this call being made spontaneously during play? Normally, a player should expect to have his backstory vetted by the GM before play, and any major points of play he wants to be established in the fiction should be included in the backstory. For example, a player ought not to expect that they can insist that they are a traveler from another dimension ("Earth") or that they are a cartoon character that was animated by a powerful magic, or anything else that would be wholly and completely novel in the setting without buy in from the GM. Indeed, pretty much everything in a backstory ought to be negotiated with a GM before play. Once the backstory is established as being in fiction and part of the setting, both the GM and the player can expect to make calls using it, but GM's should be careful about trying to impose new backstory on a player against their wishes and respect their wishes if the player strongly objects. Likewise, if a player calls something new based on his backstory, the player should expect that certain calls which are inappropriate to the setting or story or which seem to be being made solely for gamist reasons (ei, to gain some mechanical advantage) might get vetoed.
If it's established that you were an orphan, it's probably a reasonable call that you were raised in an orphanage.
Is the player correct that they were raised with a boy named Franics (sic) at said orphanage?
Was it in the player's backstory prior to the beginning of play? If so they were correct. If it wasn't, they are only possibly correct. In general, if it was established that the player grew up in an orphanage, there is nothing unreasonably about claiming that you knew someone named Francis (assuming Francis is the sort of name NPC's have in the setting).
Is the player correct that Francis and the PC were very close and dear friends?
While all the above comments still apply, sure, why not?
Is the player correct that this guard looks like Francis?
Again, while all the above comments still apply, sure, why not?
Is the player correct that this guard is Francis?
Here is where things get really dicey. It's generally considered poor form to try to use your backstory to gain mechanical advantage above and beyond what is written on your character sheet. 5e D&D has no built in "contacts"/"circles"/"allies" check and no built in way to list such things as preexisting in the setting. In a game that did have such things, "Francis" would need to be written down in a column somewhere which had a finite number of called out allies, and a suitable description establishing that they were a guard in a particular location. In that case, the player by calling out "Francis" from his character sheet would be doing something similar to calling out the rope in his backpack that was part of the preestablished fiction. The player would have some mechanical device for negotiating with the GM regarding the narrative and establishing the truth of something in the fiction. He might perhaps get a "circle test", and might have some reduced difficulty of some sort because Francis was a known established resource. Then the fortune mechanics of the game would establish whether this was indeed Francis in a way that everyone had agreed was fair and reasonable prior to play.
None of this is true of 5e. There is no mechanics available to the player for negotiating what is in the setting. This means that the situation has to be resolved by fiat, and in D&D, only the GM has fiat authority. Players can't establish things by fiat. They can only propose things that they want their character to try to do. The general rule about this is, "Could you as a real person cause someone to be someone you wanted them to be merely by wanting it to be so?" No, you can't imagine the way you want reality to work, and therefore make it so. Since normal people can't simply alter reality with wishes, your character needs some sort of explicit power or resource that they can call upon to alter reality. Essentially, they need some sort of packetized narrative force (like a spell or power). No such power exists in D&D so far as I know, short of something like spending a Wish.
So chances are, the player is NOT correct this is Francis. The player can make a call like, "Is this guard Francis?", but the GM has no way of deciding that in D&D except by fiat, so he has to make a ruling. Since rulings are outside the written rules, it's entirely up to the GM how to handle this and none of the ways are wrong. He might say "Yes." He might say "No." He might give a flat percentage chance that it is so? (If that is the case, in some games the player might have some power of Luck that modifies random rolls, and that might be applicable.) Or he might invent some sort of test on the spot that seems good to the GM. But while you can propose, "Is this guard Francis?", you can no more make it so than you can propose, "I jump over the Ocean in a single bound." Less, because the second is an action, while the first is simply a question.
Imagine the consequences of violating this simple and obvious interpretation of the process of play. If a PC can propose, "This guard is Francis.", can they also propose, "This chest contains 10,000 gold pieces?" Can they propose, "I once saved this Red Dragon's life by healing it of Dragon Pox." Are you seriously advocating for a process of play where every statement a player makes about the environment is a statement of fact? Such a process of play might be suitable for Toon - but even Toon has the rule "only if it is funny" - but probably not for a game intended to be serious.
Is the player correct in that Francis the Guard still thinks of them as a friend and wants to help them out?
Depends on what has been established about Francis before this moment of play. The player could be correct that Francis is Guard still thinks of them as a friend and wants to help them out, but that has no bearing over whether this is Francis and he is here right at this moment. That's the thing that is really at stake.
What do we do if the player then insists, "But I know Francis is a guard in this town, we had drinks before I left on my grand adventure."
Well, again, that depends on what has been established prior to this moment of play. The player might well be correct about that, but that doesn't establish that this is Francis right now at this moment. The GM isn't obligated to even say that this guard looks like Francis. He might say, "No, the guards at the gate are orcs that look nothing like Francis." What has happened to Francis, might be an interesting thing to resolve during play, but it's not up to the player to decide the answer to that - only to uncover what that answer is, if they can.
Is the PC delusional or does Francis the Guard exist? IF we can never tell the player that they cannot know something, because we cannot tell them what to think, how do we resolve this?
You're creating a false dilemma. A player is always free to establish that the PC is delusional, and if he insists on something false to facts regarding the setting, then the player is making the claim that his PC is delusional. A player could decide that, all facts to the contrary, the PC believes this is Francis. That is the player's prerogative. But the player cannot establish the facts of the setting except as provided for by the process of play. As a GM, I would be perfectly happy telling the player, "This is not Francis." That's a statement of fact. I might be happy telling them, "You aren't certain if this Francis or not.", depending on whether I think the player could tell if this is Francis. But I really can't tell the player, "Your character doesn't believe that this is Francis." if they want to insist that the character believes that it is. That highly unusual stance might require some negotiation so that I understand what the player intends, but again, if the player insists the character has false to facts beliefs and the player understands that they are false to facts, I'm not going to overrule them and tell them to play otherwise. Presumably the player has a good reason of their own for playing that way, and I'll try to facilitate that role play.
Is it not okay to tell them what they think, but it is okay to tell them they are delusional and unable to tell reality from fiction? That seems to be a pretty major thing to force upon a player.
This is where the whole statement gets ridiculous and turned on its head. I didn't tell the player that their character is delusional and unable to tell reality from fiction. The player told me that. I didn't force anything on the player. If I tell the player that a wall is 30' high and they tell me that no, it's 3' high, and they want to step over it, then they can RP out that as they like, but the wall will be 30' high and they will only be getting over it as provided by the game's process resolution. If I tell a player that the chest contains copper coins, and they tell me that the character believes that they are gold, fine. But that assertion about the player character's internal mental state does not alchemically change copper to gold. If we are going to adopt a rule where everything the character believes is true, then the character becomes immediately more powerful than the gods in my campaign world, because that character now has the power of fiat.
