In Fringe, the lead characters essentially are The X-Files 28th anniversary 1992-2020 thank you for the memories shirt government conspiracy (recall Season 2, Episode 1 when Broyles tells the new agent to forget about what had happened at the car crash and hands her the “official report”). In addition, the conspiracy is about alternate/parallel universes, not alien colonization. From a high level, the two shows are similar: there’s some greater “truth” that the characters are involved in and are trying to decipher. However, the mythology for the two series are so completely different, it would be nonsensical to have Fringe somehow crossover/merge with The X-Files. As mentioned in the question description, I suspect it was just the writers dropping a cute pop-culture reference for the viewers to appreciate. With regards to the x-files as shown in the tv show yes they do exist but they are not really everyday occurrences in the life of a few people as shown in the show.
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There are a couple of hundred cases which have been The X-Files 28th anniversary 1992-2020 thank you for the memories shirt you couple that up with the fact that FBI has under its aegis millions of people.. those cases are just an unexplained rarity which may or may not be just psychological experiences of a few people. The series was narrative fiction which used the framework of real-life events and speculative anecdotes to construct its stories. The interiors of these structures were filled with fictional people and events which form the episode’s cores. None of the events which occurred in the series happened in real life. Think about it for a second: If anything that happened in the series could be confirmed to have happened in real life, then agents like either Mulder or Scully (the protagonists) could easily quit their roles in the FBI.