Is it poliwag then? Poliwag barely has legs, has thin, transparent skin, and the line says "shall", not "can", implying that it isn't poliwrath, but maybe could be in the future. Plus, poliwrath can swim through stormy seas.
You got it right. There are actually a number of hints that says this is not Poliwrath.
I wish my legs were stronger. - Poliwag has weak legs but makes up for it with it's strong swimming skills. This is made much more clearer in it's Pokedex entry than it is for Poliwrath.
My skin is honestly too thin. - Poliwag has thin visible skin where you can see it's intestines.
But the doc is proud of my power! - Doc is a reference to "Dr. Bug" which is the nickname of Satoshi Tajiri, the creator of Pokemon, whose favorite Pokemon of the Kanto region is Poliwag. This is another clue that hints that this is not Poliwrath.
And I'm proud of my power! - This is a reference to one of Red's Pokemon of the Pokemon Adventures manga, Poliwrath who at one time was a Poliwag. His characteristic is that "it's proud of it's power."
I'll brave through the storms! - There are actually two clues here. For Pokemon Adventure fans, this is a subtle hint to the fact that Red's Poliwag and later Poliwrath has a Brave Nature. To video game fans, this is a subtle hint that Poliwag has the ability Swift Swim.
Withstand against the most powerful move this Stadium offers. - This is a reference to Pokemon Stadium's description of the move, Explosion, where it's described as "The most powerful attack of all." which holds true at the time of Gen 1 and Gen 2. Unfortunately the "most powerful move this Stadium offers" is no match for Poliwag's Damp ability.
And one day no mind shall escape my sight! - This is another clue where it hints that this isn't Poliwrath as it's talking about the future. For those not aware, Poliwrath has access to the move "Mind Reader" but not Poliwag or Poliwhirl.
Everything will be okay. Because I am here! - A continuation of the last line, this could be perceived as still talking about Poliwrath in the future. For those not aware, this is a quote from the anime, My Hero Academia, which references Poliwrath's superhero-like nature from the Pokemon Adventures manga, saving Red from almost certain danger not once, not twice but three times.