I cannot speak to the experience of any other city apart from New York, but to answer your question for New York: It depends, though I cannot think of anywhere named 'Union' specifically for a labor union. Landmarks or streets named 'Union' are either named for the United States, or in honor of something entirely different.

Union Square in Manhattan, for example, is not named Union Square because it commemorates either the federal government or labor unions, but because it is where Broadway and Bowery (now 4th Ave at this stretch) intersected. That is, the square is at the 'union' of the two roads.

In Brooklyn, however, Union Street is named for the United States, but it was named before the Civil War. In the area of Brooklyn west of Prospect Park, Union Street runs from the similarly patriotically named Columbia Street along the waterfront to end at what was then just Prospect Park but what is now Grand Army Plaza, a monument to the Union victory in the Civil War. The patriotic and federal naming is common in this area, as Union Street runs parallel to President Street and Sackett Street, named after a New York congressman known for his vocal opposition to the accession of slave states into the union.

Sources: Archives of the Brooklyn Daily Eagle, maps from the antebellum period, the book 'Gotham' by Mike Wallace and Edwin G. Burrows, as well as the website of the NYC Parks Department.

I am not sure if there is any way to answer this systematically. "Union" can mean a lot of things in cultural imagination and in geopolitics, of which the side in the Civil War and organized labor are only two.

However, it is also true that the Civil War casts a massive shadow over that American cultural memory, including in street names. Although Washington is the most popular human-derived street name in the U.S., according to the Census Bureau, the most popular name overall in Virginia is Lee.

Unfortunately, the CB will only release a dataset from 1993 of the top 100 or so street names, so it's hard to get a full look. However, I tried both Autofill on Google Maps and looked at some street directories of individual counties and cities. In Southern/ish states like Missouri and Tennessee, you'll find a lot of Confederacy Circles and Confederate Drives partnered with Union Boulevards and Union Ways. There aren't a lot of Management Lanes out there.

In the UK the name 'union street' has historically been used literally to refer to the connecting functionality of roads. For example in Plymouth a Union Street connects the city centre to the docklands.