Washington repeatedly talks about the United States, in a variety of terms like "the country" or "the nation." That's not unusual for an American president—you'll still hear a lot of that kind of talk today.

And you'll hear it for similar reasons, too. Washington was president when the government first divided into parties and when the different regions of the United States didn't see eye to eye on a lot of things. Emphasizing the country as a unit is a way for Washington to motivate his audience to focus on the larger country rather than their local areas.

Sometimes he refers to the country in a very complimentary way, like when he references "the deep acknowledgment of that debt of gratitude which I owe to my beloved country" (6.1).

Sometimes it's more obviously a way to inspire people, like when he hopes that "your union and brotherly affection may be perpetual; that the free Constitution, which is the work of your hands, may be sacredly maintained" (6.3), or when he reminds them that:

The name of American, which belongs to you in your national capacity, must always exalt the just pride of patriotism more than any appellation derived from local discriminations. (10.3)

Even when reminiscing, he emphasizes the United States as a nation, like when he mentions his neutrality proclamation of 1793:

I was well satisfied that our country, under all the circumstances of the case, had a right to take, and was bound in duty and interest to take, a neutral position. (45.1)

When Washington made that proclamation, a lot of people were mad at him for making the proclamation on his own, without Congress. Three years later, he paints it as the policy of the whole country.

Presidents generally use this tactic, of framing everything in relation to the nation. (Which would also be a great song title—but be sure to credit us.)

This strategy makes it look like the politician is thinking about the country above all else and that all the politician's decisions are about the country, rather than themselves or their buddies. It also potentially gets people listening or reading to think about the country, too.