I read the point as being: spray Rhydon with bubbles and water? Faints. But ride it across the ocean, which is made of water? Why not?
It'd be like, if you were allergic to bees, and you jumped into a hive of bees in order to spray bees at your opponent for an attack.
That was a terrible analogy. I'm sorry.
There may be a difference between using water as an attack and just swimming in water.
When used as an attack the water is being sprayed at high velocity letting it pierce/erode through the Pokemon's body and causing a negative reaction with the Pokemon's Type.
But when just standing/floating in water the Pokemon's outer defenses is enough to prevent the water from causing any pain or discomfort.
And there's maybe something to this idea as let's take a look at the other Water-type HMs: Waterfall, Dive, and Whirlpool:
Waterfall: Allows a Pokemon to swim up a rushing waterfall. The Pokemon who learn it via HM/TM are either Water-type, live in the water, or at the very least wouldn't be bothered by water.
Dive: Allows the Pokemon to swim deep underwater. Once again, only learned by Water-types, aquatic Pokemon, or at very least aren't bothered by water.
Whirlpool: Allows a Pokemon to swim across a fast swirling water vortex. Now this is where things get interesting as some of the Surf oddities are here: Nidoking, Nidoqueen, Lickitung family, Rhydon & Rhyperior, Tauros, Snorlax, Sentret family, Sneasel family, Miltank, Tyranitar, Zigzagoon family, Exploud, Makuhita family, Aggron, & Rampardos. Hmm, it could be that, instead of trying to fight through the strong current, the Pokemon instead swims with the current and uses the velocity it builds up to slingshot it and the riding trainer over to the other side.
But the anime also had an entire episode focused around a Rhydon that surfed across water, so...
The anime can be confusing sometimes though. In the famous Pewter Gym battle episode Onix got massively weakened by the Gym's sprinkler system going off. But then during the Cinnabar Island episode where they were trying to prevent it from erupting you can see both Geodude and Onix lining up to "cool off" via Squirtle and Staryu spraying them with water as they helped throw rocks into the magma chamber. In the Sinnoh series the episode that debut Probopass had a plot point where Probopass and its trainer fell into a lake, something which they treated as very bad but when it happened Probopass didn't seem that much hurt from it. Finally there's been a few hot springs and beach episodes where you'll often get a shot of the Pokemon weak to water hanging out off to the side while the other Pokemon play in the water... though in a recent one we say Kiawe's Charizard perfectly enjoying itself in a hot spring. So the anime honestly flip flops whether standing water is still harmful/uncomfortable to Pokemon weak to water or they can be in it perfectly fine, so can't really take the anime's word on things in this case.