It's a UN body, so there's no strict enforcement. If the USA decides to ignore the UPU's fees and regulations, there's not much reason for other countries to accept American postage stamps and post in general under UPU rules, though.
Without the UPU, the USA would need to negotiate rules and fees with every other country on the planet individually, causing an explosion in complexity, administrative overhead, and cost. If the negotiated rates are good enough there can be a net benefit for the American government, but I doubt they'll be able to make it work.
I don't think the USA will leave the UPU, though, they're probably working up a small fine to try to force renegotiations. A dangerous game, given how sick the world is getting of America already.
During Trump's last presidency, the UPU was already involved during the trade war started against China, and the USA paid off the UPU for 7 million dollars.
