A Smithsonian article about goldfish starts out with (emphasis added):
The 2003 Pixar film Finding Nemo popularized the misconception that all drains lead to the ocean, encouraging young fans to flush their pet fish down the toilet in an ill-advised bid for freedom. As experts were quick to point out following the movie’s release, flushed fish typically die long before they reach the ocean, going into shock upon immersion in the toilet’s cold water, succumbing to the noxious chemicals found in the sewage system, or—if they make it this far—finding themselves eliminated at a water treatment plant.
I don't remember Finding Nemo encouraging viewers to "flush fish down the toilet."1 If anything, Finding Nemo (and its sequel) positively portray fish. (Interestingly, the film may be harmful for the environment.) Therefore, I am skeptical that anything bolded in the Smithsonian quote is actually true or if the Smithsonian is presenting a popular myth as fact.
Did Finding Nemo popularize "the misconception that all drains lead to the ocean?"
Were more children flushing pet fish down the toilet after Finding Nemo?
While the claim in question 2 is not directly stated, it is strongly implied. Finding Nemo "encourag[ed] young fans to flush their pet fish down the toilet." If Finding Nemo did encourage this behavior, this behavior would increase after the release of Finding Nemo. I have phrased the question to be more answerable than the original claim while still mainting the same meaning. Anecdotally, the LA Times published a 2003 story about plumbers receiving an increasing number of calls for fish in toilets. GRID-Arendal also wrote about the subject without specific sources. "After hundreds of children ‘liberated’ their pet fish down the toilet, following the launch of ‘Finding Nemo’ in the United States, two environmental organisations are keen to avert a similar catastrophe when the animated block buster opens in the UK on 3rd October 2003."
1 As @F1Krazy correctly points out:
"All drains lead to the ocean" is pretty much a verbatim quote from the movie. Whether it "popularised" that myth is probably harder to figure out.