Q: “Do levels exist in the anime?”
A: No. Numerical levels are not implemented into the anime. In spite of the fact that the anime is based on the games, it is a non-game adaptation, and featuring numerical statistics that equate to battle experience would simply be unrealistic. There is some discrepancy in regards to what a “level” means in the anime, as it is occasionally brought up in dialogue. Generally, this refers to levels of power or experience, not a digital number that somehow exists in the confines of a more realistic setting. But Pokemon do not evolve or learn new moves based on a literal numeric level. Not to mention the series features various inconsistencies in regards to a Pokemon's power.
The most common rebuttal to the argument of nonexistent levels is Episode 9, “The School of Hard Knocks”. In the episode, numeric Pokemon levels were repeatedly mentioned similar to the games. Mind you, this episode was very early in the series, and over a decade has passed without a mere mention of such. In comic book terms, the concept was “ret-conned”, in which continuity is retroactively changed. This is not uncommon in a continuing series, especially when you consider how back in the show’s earliest episodes, the writers were still working out the kinks in the story and deciding what worked and what didn’t.