While not OP, Stamina is still a really good ability. Especially in-game, if you can accumulate a few Defence boosts you can plough through your opposition with little trouble. On top of that, a lot of the Pokémon you'll fight in Alola (kahunas and Totems and whatnot) are physical attackers, meaning that Stamina will see a lot of boosts. However, Stamina is not overpowered. Powerful, yes, but it does have weaknesses.
- It only raises Defence. Mudbray and Mudsdale don't have very good Special Defence. Much like Fluffy Bewear, the solution is to attack with Special moves.
- Mudbray and Mudsdale are pure Ground-types, meaning their weaknesses are Water, Grass and Ice. All three of these are mostly special. There are very few commonly-used physical Grass moves (Leaf Blade, mostly) and Ice moves (Ice Shard and Icicle Crash), and a lot of Water Pokémon rely on Scald and Hydro Pump as their main STAB moves. This presents problems for Mudbray/Mudsdale, as Stamina won't usually save them from attack that they're weak against.
- Stamina is less useful competitively than in-game for a few reasons. First, you can't burst-heal with Potions like you can in-game. Mudsdale has to take damage, then gain the Defense boost, then take more damage. Not to say that Stamina won't be good, just that you can't accumulate boosts and then use a Hyper Potion. Mudsdale also has no way to heal itself outside of Rest, Sitrus Berry and Leftovers. Second, in competitive Pokémon tend to switch a lot, and switching removes the Defence boosts — meaning that an awesome boost to Defence can be removed just by having your opponent bring in a threat to Mudsdale.
In summary, Stamina is not OP. It is a great ability, and once you have a few Defense boosts Mudsdale becomes nigh unstoppable with physical attacks, but it does have a few flaws that prevent it from being ridiculously overpowered.
Now if it raised both Defence and Special Defence…