What are some noun and adjective that can describe a person or a group of person who think themselves elites, but I don't necessarily agree?
To indicate that someone is claiming a title or stature that you don't think they deserve, you can refer to them as self-appointed or so-called; you can also put their label in quotes to indicate that you disagree with it:
These self-appointed "elites" seem to believe that they know better than you do how to spend your money.
These so-called "elites" can't even tell a salad fork from a dinner fork.
It depends on whether you're calling into question the elite status of the group to which they claim to belong, or whether you are calling into question whether they really belong to a group you consider elite.
For instance, if you wish to disparage people who think their superior poetry-writing skill makes them superior to those people who aren't into poetry, you might say they are "putting on airs", "full of themselves", or "snobs". If, contrariwise, you wish to disparage people who think their poetry-writing skill is enough to be considered an elite poet, you might call them a "wannabe" or a "poser".
For people that act like they are better than you and are rude about it, I personally like to use...
If you want to use less common, more formal words for this, there are dozens, including...
Some other honorable mentions after looking at the thesaurus are...
There are so many good words to describe these kinds of people. Take your pick. :)
I'd also add, if you want one word you can prefix the name of the group with "pseudo" or "faux", both meaning "fake". Like, "Bob and Tom think they're smart but they're just pseudo-intellectuals." "Sally wants everyone to think she's very creative but really she's just a faux poet."