Shocking!
Why do chess.com players put up fake pictures of themselves as their profile picture?

It's fine putting up a photo that obviously isn't you - I am not a shoe - or a photo of a celebrity. But a photo of an ordinary person does seem odd.
This is exactly my point. What are these people up to?

Think I have an identity problem.
Cool name. So you are a toon or whatever Sunderland people are called? (Non-Brits relax for a few minutes.)

You need educating #31 Sunderland are "Makums" and heading south. Check out your geography for 55 - it's Newcastle all the way and heading up - it's Toon Army - but thanks for the chat

Ooh, hurt feelings? Passive / aggressive or what? Info for non-Brits, Newcastle and Sunderland are identical. I love Newcastle. I had one of my best Saturday nights ever there, but I don't remember much if it. Thanks to my friends for getting me home.
If there is anyone here who wants to come to England but doesn't want to go to London, then go to Neecastle. It's a big friendly city and they sure like to party (in my experience).

Postafi - are you stalking me? Why are you remembering what my pics were? Shivering.
"Us, mentally disturbed are overrepresented among chess players"
- @Postafi 2018

I think that "fake", or " altered", pictures are dishonest (or at least misleading) if you claim to be that individual in the avatar. However, an avatar of a celebrity (or famous chess player, or any other well known person for that matter) does not really fit into this category, because the owner of the account is not claiming to be that person in the avatar. Why would anyone "act" as someone else? Perhaps due to personal insecurity, maybe to conceal their own identity, or many other reasons. However, if you really want to protect your identity - that is no reason to impersonate someone else. Your avatar could be a landscape, a pet, a TV character, and many other options.
I think that everyone should be aware that chess.com "friends" is indeed a friendship (potentially strong ones too), but it is not even close to knowing that person IN PERSON. There is always the possibility that the avatar is not a photo of the account creator, but this possibility also adds a little security if your avatar is indeed you.
Personally, I find some of the best avatars to be creative; this is seldom a photo of a person (yourself or not). Of course, whatever avatar is used is really up to the account creator - and is up to them what they choose as an avatar (chess.com permitting of course, as it is ultimately their site).
Let us not deny that many Chess players are mentally ill but not dangerous, though.