What It Means to 'Square the Match'
(Before we continue: What is "match play"? "Stroke play" means that you count each stroke you play, and at the end of your round you add up those strokes, and the resulting number determines whether you win or lose or where you stand relative to other golfers. "Match play" means that golfers attempt to win each individual hole, and the golfer who wins the most holes during the match is the winner of a match. If you win five holes and I win four holes, then you beat me by a score of 1-up (and I lost by the score of 1-down).)
So "square the match" is a term used only in match play, because it means that the trailing golfer won a hole and now the match is tied. If I am 1-down (you've won one hole more than me) after 15 holes, and then I win the 16th hole, I just "squared the match."
On a broadcast of a match-play tournament (say, the U.S. Women's Amateur Championship), an announcer might say, "She can square the match by making this putt." (Which means if the golfer sinks that putt, she wins the hole, bringing the overall match into a tie score.)
"Square the match" derives from the very similar term "all square," which simply means "tied." For decades, the rulemaking bodies of golf, the R&A and the USGA, only used the term "all square," and never used the word "tied," in the Official Rules of Golf and in statements about match play. Then around the year 2020, they decided to lighten up, perhaps one day thinking, "why not use a word that everyone understands, instead of a term that only golfers understand?"
Since then, golf broadcasters, journalists, bloggers, pro golfers and recreational golfers alike, have begun saying "tied the match" in addition to "squared the match." Over time, we should probably expect "all square" and "square the match" to fade from common usage. But that will probably take some time, since those were the preferrred terms (the only terms used by the USGA and R&A) for a very long time.
More definitions:
Sources:Davies, Peter. The Historical Dictionary of Golfing Terms (affiliate link), 1993, Robson Books.
R&A and USGA. The Official Rules of Golf. https://www.randa.org/en/rog/the-rules-of-golf