Why is this not tied to a person's SSN (if possible)?
The point of that database is to display a recognizable name to the people you call, so that they know it's you. A recognizable name isn't always the one on your birth certificate (particularly in the US). There are also businesses, who want their business name there.
Also, many postpaid plans (like my home ISP) require SSN because they are providing you service on credit. Postpaid cell paone plans have been the "default" in the US for a long time, though prepaid seems to be gaining market share.
They require a SSN because people don't care and it makes it cheaper to offer the accounts, not because it would actually be a big problem to sell internet service without credit checks.
Now, as for why they still do the same credit checks when you bring your own phone, I suspect "Because F you, that's why" is the gist of it.
As Brit-expat+US-resident (since 2012) T-Mobile got my SSN when I signed-up for my pre-paid first mobile phone plan in 2012. Paying $50/mo was quite a shock when equivalent (or rather: far superior) service was available in the UK on a PAYG (not even pre-paid!) basis for £10/mo.
...and now I'm on a $110/mo postpaid plan because eventually you get tired of the limitations and just grin-and-bear-it.