It's also a way to flex. Same way people tend to wear alumni sweaters from Ivy League, Stanford level schools. I doubt you'd see anyone wearing an Infosys (or similarly viewed company) shirt in that kind of environment.
Right, but no matter what your past is, you are still at $currentCompany, not $prevCompany, so in that most important of ways you are exactly the same as all these other people you are trying to flex on. Why would I care that you used to get paychecks from a more prestigious company if yours are signed by the same guy as me now?