It’s definitely an old school term dating back to when people made their own butter. Here is a site that talks about making butter and uses the term.
Link is here.
What was left was made into nice round pats, crossed on top with the butter pat, and this was used for our table.
And here is another site talking about a butter mold, and they mention the “pat size” (the molds are at the bottom of the page).
Link is here.
The third, fourth and fifth molds are fancy carved butter molds. The first is a one pound, the second a half pound and the last one was a pat size
I had never heard this term before until I went into a local country store and they had a sign up selling “pats” of butter which were round, 1/2 pound blocks wrapped in paper. It seems to be a rather loose term as I can’t find many references to it.