The best racing games like Wave Race 64 are Wave Race: Blue Storm, Riptide GP: Renegade and Hydro Thunder Hurricane. Below are 7, each with the reason it is here — and, at the end, what filtering to racing costs you.
Filtering to racing means you want the vehicle to be the subject. It is the right filter when what you liked was speed, or handling, or a road worth driving.
What you get: A short session that is complete in itself. A race is over in five minutes and you can always take the corner better.
What made Wave Race 64 worth playing in the first place is covered on the full list. This page is the racing part of that answer, and only the racing part.
The GameCube sequel, and the only real continuation.
The closest modern game — jet skis, waves, trick system.
Arcade boat racing with big water and bigger jumps.
By the Riptide developer, and directly modelled on Wave Race.
Arcade racing that keeps the light tone.
If the N64 arcade racing feel was the appeal.
For the same obsession with mastering one course.
Everything outside the vehicle. Racing games rarely have a story, and the ones that try usually should not have.
Specifically, these are strong answers for Wave Race 64 that are not racing games, so they are not on this page:
The full list is on games like Wave Race 64 — 9 picks, no genre filter.
Wave Race: Blue Storm. The GameCube sequel, and the only real continuation.
No. Wave Race 64 released on the Nintendo 64, but a recommendation is only useful if you can play it, so the picks above span platforms and eras. What they share is the genre and the reason.
Simulation models tyre temperature, fuel and weight transfer and punishes you for ignoring them. Arcade gives you grip, boost and a rewind, and is about flow. Buying the wrong one is where most disappointment comes from.
For the simulation picks it changes the game completely. For anything arcade it makes almost no difference.
By hand. Each pick was written for the full Wave Race 64 list first, on what the game actually does, and this page is the part of that list that is racing — never a genre tag lookup, which is how "best racing games like Super Mario 64" used to be a page on this site.
Or read the racing guide for where to start in the genre generally.