Outside of maybe some unusual gamemode, Dota 2 doesn't allow you to change hero midgame. Overwatch is a hero shooter, not a MOBA, so the mechanics of that genre aren't necessarily good things. This would also be a massive game change that requires an entire rebalance of the game, and may not even be feasible to handle from a code level. It also removes a lot of the purpose of pick-ban: being locked into what you're playing for the game is what makes the phase so impactful and what gives things like counterpicks and scaling picks their meaning.
Imagine, for a moment, playing Renekton top lane (or some equally strong early game top laner). you get a lead, maybe get a couple kills with jungle intervation, but you start to fall off late game. Except now you just switch to Kayle, and you get to skip her entire weak laning phase and difficulty farming to get her god tier late game earlier than most Kayles could due to their early game weakness, and still maintain your advantage over the enemy team.
Being able to switch while in base in overwatch works because the hero kits are all there is, there's no added bonuses outside of them. Thus, switching is simply a tactical decision to either counter an enemy choice of pick something more suited to the situation. In league, with scaling, snowballing, and items, that system wouldn't work.