You’d think that, but chess has two rule against this: if a sequence of moves is repeated three times the players may declare the game a draw, and if there’s 50 moves played without captures or without pawns moving, the players may declare the game a draw. There is no indefinite chess game possible with these rules.
There are indefinite Go games possible depending on the rule set (you’d need a superko rule to prevent infinite games due to things like triple ko). However, in every ruleset it is possible to play games that last more moves than there are atoms in the universe, if both players wish to do so.
Even if one of the players does want to end the game, the other player is usually able to stretch a game that ought to be over in about 300 moves to a game that lasts about 600: one can waste stones by throwing them in the opponent’s territory, one can fill up one’s own territory, one can even let their own stones be captured, and if the opponent is naive enough to capture, you got more enemy territory to waste moves in.
This kind of stalling has nothing to do with the game of Go, and is infuriating to opponents. It prevents them from ending the game short of resigning, while they are actually ahead in score, and nothing the opponent does will change that (it couldn’t even, since at this stage of the game territories are unconditionally alive, and the winning player cannot be killed anymore).