Phone service: random UID as you suggest, users put aliases in address book. Works great. Plays no favorites, unless you want 555-555-5555.
Internet DNS: restrictions on valid names (to certain TLDs) which are gradually lifted as our name needs increase. Expiration, which favors the needy. Cash market, which favors the rich and needy.
Email: Similar to DNS, but mandatory attachment to domains broadens set of available names. No expiration policy at high demand domains (i.e. gmail).
And if you're referencing me from your phone, you could alias me as @knowsnothing and change my icon to a koala if you wanted to. That offends me less than me having to come up with @awesome_sauce314 to use some photo sharing service, which I'd need to share among friends, coworkers, family, strangers, etc.
How will someone born today have any hope of any kind of usable identity in a system that allows only one instance of a name.
Managing overlapping names among friends is something most people know how to do well enough already.