Because he doesn't need to.
In the games, the in-universe reason for catching every type of Pokemon is to complete the Pokedex, to help the regional professor with their research (the out-of-universe reason is, of course, sheer completionism). The Pokedex does not contain complete information on a Pokemon species until you catch at least one Pokemon of that species. This is the same reason that Red, in Pokemon Origins, catches all 150 - because Oak asked him to, for research purposes.
In the anime, the Pokedex is already complete. Every single time Ash encounters a new Pokemon (with a few exceptions), the Pokedex lights up, shows its picture, and rattles off a short expository explanation. There's no reason for Ash to go around capturing one of every Pokemon, because neither he, nor Oak, nor anyone else I can think of in the anime, needs one of every Pokemon.
Doesn't becoming a Pokémon master mean catching them all?
No. I'm not sure whether the anime actually has a defined criteria for what constitutes a Pokemon Master, or whether it's just some vaguely-defined goal for the writers to dangle in front of Ash so they can justify the series going on and on and on. [I'll look it up later.] My best guess would be that to become a Pokemon Master, you have to have won a regional league, something Ash has systematically tried and failed to do in every single region so far (other than the Orange Islands, which apparently don't count).
The point is, in neither the games nor the anime does capturing all (now 800+) Pokemon make you a Pokemon Master. In the games, all you get is a lousy Diploma and a sense of completionism. In the anime, it's a completely impractical and pointless goal that, in spite of the series' tagline, isn't really worth pursuing at all.