Not so much. The "flair" I've seen is very, very limited (so a bunch of people will end up with the exact same ones; there might only be a handful of choices). And part of the usefulness of Reddit is having the same user account across the entire site, so you don't have to separately log into a dozen different forums about different topics. A creative username for a car-related subreddit is going to be pretty useless when you browse over to some subreddit about programming, where no one cares about your obsession with Skylines or whatever. Finally, that stuff isn't really visual the way iconic avatars are, where someone has some small picture for their user account.
There are custom text flairs available on some subreddits (for, say, camera models in r/photography) as well as icons (the club badges in r/soccer). It's quite flexible.
Reddit now has profile images too but they are only displayed there.
Most of the subreddits I frequent allow you to choose freeform flair, and it's specific to each subreddit. Sure, though, it's limited in length and it's more closely analogous to a signature than an avatar.
