I own a Nintendo Switch, and I've noticed that in the Nintendo store, old games regularly go on sale for in the ballpark of 80% off. Does that happen in the PS store?

They do run sales, but some of the good sales are gated behind paid online membership. Some of those gated sales require a more premium membership plan.

Many games seemingly never go on sale, while others seemingly are always on sale.

third parties do. Good luck buying a nintendo game for less than it was at launch

Nintendo occasionally does 30% off sales on their published games through the eShop. There are a few active now: https://www.nintendo.com/us/store/sales-and-deals/

A funny quirk of that is that the Mario and Rabbids games are always 90% off because they are not published by Nintendo.