Once upon a time, the freight railroads also operated passenger service, and initially each railroad typically built its own passenger station, which was named after the railroad. Eventually, some of the railroads decided to work together to build stations served by multiple railroads, which is how the name Union Station came about.
Were some of the Union Stations also named because they were built by the Union Pacific?
Union Station in Salt Lake City, which is now an entrance to the Gateway Mall, has a big Union Pacific logo on its roof. Just down the street is a beautiful station with the Rio Grande's logo on the roof.
I'm guessing this particular Union Station is "Union" because of the Union Pacific, not a union of disparate railroads.
Perhaps LAUPT is the same--though it may have been served by the SF (or AT&SF or whatever it was in those days), SP, UP, and maybe another company or two, so it could have been a "union" of railroads which coincidentally included the Union Pacific!