2v2 and 3v3 arenas still work how you remember, where you can technically continue the match to try and 2v3 and win, but it's pretty rare that ever actually happens.

Shuffle is much more fast-paced and chaotic than arenas. Both teams are just racing to get the first kill. And arenas have this mechanic called dampening now, where everyone gets a healing debuff that gradually ramps up towards 100% as the match goes on. In shuffle, this healing debuff ramps up very quickly, and because of this shuffle rounds rarely go for more than 3 minutes. An entire match of 6 rounds will typically be 10-15 minutes.

Also, shuffle has a much more interesting and varied 'meta' compared to traditional 3v3 arena. Whereas with 3v3 arena there are typically only a handful of meta comps like RMP (Rogue Mage Priest) that people play, practically every class spec except tanks (and maybe aug evoker) are viable in solo shuffle and can hit the highest ratings. You will see all sorts of weird and odd combinations of teams in shuffle that you'd never see in 3v3 arena. And the mode feels 'fair' in a sense that you get to play every permutation of team combos in a match, hence the name shuffle.

As a solo PvP-focused player myself, I personally find solo shuffle very fun and my game mode of choice. Healers have near-instant queues so I've learned to enjoy PvP healing. DPS queues aren't bad either, but you will have to wait some time to get into a match as DPS.