Washington uses the term "union" a lot in this address—13 times, to be exact. Similar to some of his other rhetorical devices, he's really pushing the "let's be one big happy family" theme. It's not so surprising given that his Cabinet saw the rise of political division and party politics in the brand-new nation.

Sometimes he uses the uncapitalized version, like when he reminds the audience that "it is of infinite moment that you should properly estimate the immense value of your national union to your collective and individual happiness" (9.3) or about "powerful and obvious motives to union, affecting all parts of our country" (14.7). This is when he's talking about "union" as a broader concept, likely something more aspirational.

There's also the version that's written with a capital letter. For instance, he talks about "the continuance of the Union as a primary object of patriotic desire" (14.1) and ponders "the causes which may disturb our Union" (15.1). In these cases, "Union" is a sort of synonym for America, but one that emphasizes the unification.

Kind of like the United States of America.

Remember that the United States started as 13 colonies, and before there was a Constitution, the colonies were held together very loosely by the Articles of Confederation. Getting people to think of America as one country wasn't easy, so it's not that big of a surprise that Washington would really try to get people to focus on the bigger picture.