Are you meant to be "grown up" at 21?

That would depend on what your definition of a 'grown up' is?

If we're talking about maturity, I'd hazard a guess and steer it in the late 20's or 30+ category. I'd like to think that your twenties are for discovering or even rediscovering yourself; the following quote sums it up for me really.

“Your 20’s are your ‘selfish’ years. It’s a decade to immerse yourself in every single thing possible. Be selfish with your time, and all the aspects of you. Tinker with ****, travel, explore, love a lot, love a little, and never touch the ground.”

Really depends who you speak to. Ask someone in their 80s and they'd tell you that in their day 21 year olds were getting married, fighting in wars, raising families etc. So yes, definitely.
As someone in their 20s now and they'd say no, because these days even people in their 30s a lot of the time are behaving and being treated like teenagers.
It's the paradigm shift.

I got married at 17, owned my own home at 18 and moved in with my fiance just before I was 19, and got married and had my first child at 21. So yeah, to me 21 was grown up.

Some other 21 year olds are less mature than my 9 year old kid though.

Swings and roundabouts.

I understand what you're saying with this and I don't think you mean anything else by it other than what you've written and for you obviously having kids and getting married was the stage in your life where you felt grown up. But, I personally don't think that marriage and kids is the definition of 'grown up' for everyone - it is to some people but I know plenty of very immature people from school who had kids as teenagers and married in their 20s and they still act like children themselves. Maybe they feel like adults, I don't know, but they don't project that image other than the fact that they've done 'adult things'.

I'm a lot more mature and responsible as a childless mid twenties graduate than people I know from my school who don't work or do anything in particular but are married with children and have a house of their own (not mortgaged but not living in a bedsit or with their parents). So yeah, I don't think doing certain things makes someone an adult, I think it's the way they behave, their sense of responsibility for themselves and their outlook on life.

Well when you have to pay the bills and take care of a family, it more or less forces you into a position of responsibility. I meant as a direct contrast to people in their 20s and even 30s who have their parents washing their socks for them!

Sometimes doing certain things means you have no choice but to grow up a bit.