The thing you have to understand with the anime is that it is not exactly like the games. Think of it as an alternate version of the game, with different rules and a different storyline.
Because of this, the anime writers can change the game's rules or ways to make the show more interesting and enjoyable to watch. One example is the fact that levels don't exist in the anime, or at least are much less significant than they are in the games. In the anime, Pokemon evolve for their own reasons, not because they have reached a level that will suddenly make them transform.
This is where Meowth comes in. Has Meowth ever wanted or needed to evolve? No, he hasn't. The writers keep it this way because Meowth is more interesting to watch than Persian; Persian is a large Pokemon with a less appealing design, and also doesn't flow with Team Rocket as nicely as Meowth does. Have you noticed that the the main characters in the story carry small Pokemon? Ash's Pikachu, Misty's Togepi, Iris' Axew, and more are small because the show is more entertaining with those small Pokemon. If they evolved, they'd have to lug around huge Pokemon; Iris' Axew, for example, would not have the same charm if it was a Haxorus.
Another reason is Giovanni. He owns several Persian. Team Rocket are considered inferior to Giovanni, so it's logical that they have Meowth and Giovanni has Persian.
tl;dr: it's because Meowth doesn't want to evolve, and because the writers know that Meowth appeals to the show's audience more than Persian does.