There are a variety of possible logical fallacies here, depending on exactly what is meant and what assumptions you make about the implied audience.
If you read the meme as:
... why did Elizabeth Warren ever pretend to be...
Then the joke is attempting to poke holes in the argument that white privilege is always superior to any form of minority privilege.
The joke seems to be broadly and obliquely referring to "the Left", who probably do not hold quite such a broad interpretation of white privilege, in which case, it is a strawman.
You could also claim a no true scotsman (all Democrats believe that white privilege...) if you want to try and restrict the audience to only people who believe in such a broad definition but still say "Democrats" or "the Left".
At the very least, it's a non sequitur since the joke does not identify the audience whose belief on white privilege is necessary to know in order to know the resulting accuracy of the joke.
Finally, if you really do believe that all Democrats actually hold such a broad view of white privilege, and are willing to assume the audience as Democrats or "the Left", then it's still a claim to improper or biased authority because Elizabeth Warren could have a variety of reasons other than pure net gain for claiming American Indian ethnicity or could have misjudged the net gain even if that was her goal.
As a side note, if you are willing to narrow the joke to trying to say that Elizabeth Warren does not believe that white privilege is so broad, and that her actions prove this opinion, then you still run into an unsubstantiated assumption of motive. Warren could have a variety of reasons for claiming American Indian heritage, not necessarily one seeking an advantage. Most charitably, she might have actually thought she was American Indian, and stated so as a plain matter of (honestly incorrect) fact.
... why did Elizabeth Warren continuously pretend to be...
This is simply unfactual. I suppose it's a faulty premise, if you want to be technical.
Although Elizabeth Warren formally claimed multiple times to be an American Indian, it seems that she went through most of her life being considered "white".
just a joke
Jokes really more on inference and emotion than formal logic, so you can't call it a fallacy.