There are absolutely some breeds that don't mature until later but stock horse standards (again AQHA/APHA) are that horses are started under saddle as 2 y/o.
Generally, however, these types of horses (if fed and exercised correctly from birth) resemble their "senior" counterparts at 2.
We have a 2 y/o red roan QH filly right now at my trainer's place that I swear looks like she is 6. (but that is what good feed and LOTS of exercise will do)
Horses that are weanlings and yearlings that are kept in little small pens with not a lot of grain.... have their growth stunted. Seriously. I've seen it. We have it right now at the boarding facility I am at. There are like 3-4 2 y/o that I think look they just came off their mama's a few months ago!!!!! It's sad really... their owners think they are going to get like 16.2-17 h!!!!!!!!!! Ay, ay, ay....
And the whole mental growth thing? Well.... let's just say we don't keep toddlers out or preschool or daycare or not teach algebra or something because minds arent ready for it (mine was never ready for algebra... LOL LOL!!!)
This is where patience, going slow, and LOTS of time comes in. To make sure that you don't blow their minds (or their legs) when starting them as 2 y/o.
Stock horses are able to show in the following classes based on age:
Weanlings (halter classes only, obviously!) lol
Yearlings (halter, lunge line, and in-hand trail)
2 y/o (green horse, 2 y/o futurities and Junior)
3 y/o (green horse, 3 y/o maturities and Junior
4 y/o (Junior horse or green)
5 y/o (Senior horse)
In fact, I remember way back in the day when it used to be Senior horses were considered that way at 4.
All the terminology means is that the horse should be "seasoned."
That is trained, going in a curb bit, and VERY competitive.
I hope that sheds some light and helps!!!!