Well, we have very little idea of the Presidents of the US in the Trek universe. Perhaps the first black president was actually somebody looking suspiciously like Brock Peters, elected by a landslide in the early 1970s after the Benny Russell riots?
Also, television often sort of preempts the celebrity paradox by forcing us to assume that actors looking subtly or grossly different are actually one and the same character... And that evil twins actually are exactly identical to the good ones (save for the goatee), rather than only approximately so. Conversely, two people who look the same on screen are probably "supposedly" subtly different in "reality".
That aside, we of course have the Gabriel Bell Incident, a take on how the Celebrity paradox would play out in-universe. Which is only one step in the long ladder of such incidents in the life of the Sisko family: old Joe Sisko must have faced some awkward situations, what with looking exactly like that traitorous admiral from the 2290s; Ben's own son will end up looking like a Klingon family acquaintance; etc.
But the number of real TV shows quoted in Star Trek is actually very limited, curtailing the effect. And actors from Star Trek generally play their possible public roles wearing the identity of an actor, rather than that of a politician, a notorious criminal, a famous victim of a disaster etc. As Star Trek the TV show assuredly doesn't exist in Star Trek, this probably means that if Janeway watched television in 1996, the "rules" would supposedly ban her from spotting Patrick Stewart or LeVar Burton in any of the shows she saw, preempting the paradox.
If it were "allowed" for the actors to exist as actors, though, there would be many unfortunate ways for them to connect. After all, they are famed for their scifi roles, meaning they might (actively or passively) play a PR role in various space programs, real or imaginary (say, the TOS posse being associated with the Enterprise
rollout, or Spiner being aboard for the first DY-100 flight around the Moon in 1988), thereby allowing our Starfleet heroes to spot them when studying the past of their own organization.