The first part of this idea (want to have their own books) is a good thing. But is the second part (start collecting every book for the game) a "personal problem"?
Can be, but usually not IMO.
I've known gamers who were like this. If a game has 100 supplemental books, the gotta-have-them-all gamer collects all 100 books. And he probably lobbies to have them all included for the campaign.
I've never had a problem with a player lobbying to have "book X" introduced into a campaign. If they ask, and I don't have a good reason for saying "no", then I'll say yes. (Sometimes, a "good reason" is nothing more than "I want to play Core Rules Only".) If I
do say "no", I've never had a player press the issue - they have always trusted that I have a good reason, and they also trust that in some future campaign they'll get to play their Drow/Pirate/Ninja/Zombie/Cyborg...
However, I can certainly see the possibility that someone might take it to an extreme, and therefore become a problem.
(For me, it also helps that I almost invariably own more of the supplements for a given game than any of my players.)
Do you own all (or near all) the books available for your game? Do you buy them because you just like reading them, or do you buy them because they exist?
I own lots of games, many of which I have never played. I have two or three "games of choice", which occasionally change (the current set is WFRP 2.0, D&D 3.5 and SWSE), for which I'll buy a bunch of books.
I used to buy books for the simple enjoyment of reading them. However, a few years ago I concluded that most games run best with a limited subset of carefully-chosen supplements added, and so became much more selective in my purchases. (The trend towards very crunch-heavy books also cut down on my buying a lot - there's little to no pleasure in reading a book such as "Martial Power".)
Like, how many monster books do you have versus how many do you actually use/need?
I treat monster and adventure books as "idea mines", and so they are my most-purchased items. There's almost always something I will use, or if not then just reading the book usually gives me a bunch of ideas that I
will use.
Still, I have many more monster books than I regularly use.
(Also, adventure books in particular tend to be much more "fluff-heavy" than other books, so they tend to be more enjoyable to read... I also like (older) setting sourcebooks for the same reason.)
Do you own any class books for a class(es) that you never play?
I'm almost exclusively the DM. And I have a strict rule that I will not allow the use of any book I don't personally own (too many bad experiences to do otherwise, though I believe it's less of an issue in 4e). If I am going to allow a class book for one class, I generally prefer to allow the classbooks for all.
So, in short: yes.
When a new book for your game of choice is released, what is your thought process:
These days I ignore pretty much everything except Pathfinder and Star Wars Saga Edition * (there aren't any new Warhammer 2e books being published, AFAIK). For these books, I ask myself if the book is going to directly add to my game, or if it will be sufficiently interesting to read for the ideas alone. Only if the answer to one of these is "yes" do I buy the book.
One exception: I have a subscription to the main Pathfinder "Adventure Path" line. Therefore, I get that every month.
* There are many other fine games out there. But I'm barely gaming at the best of times, and when we do play we want to just get on with things - we don't really want to bother with Yet Another Rule System. So, fine as "Game Z" may be, I'm not currently in the market.