For the limited selection of avatars, there is the selectable avatars feature
With the generated usernames, yeah that would require a custom plugin, I think.
I guess it’s not so uncommon e.g. a music app, Auxy, has generated names for projects
A coding app for kids that I used has this feature, as well, for usernames (they also had an adjective followed by a noun, but was much broader than just animals)
In terms of scope, maybe it could start with a two word system. And you could add separate vocabulary sets for the first word and second word, through admin settings (so that the lists can be updated and edited more easily).
Maybe users could click to randomise with a new combo each time, not sure how you might want it from there
(If you would like for someone to build a custom plug-in, usually jobs like that can be posted in the marketplace if you have a budget)