Growing up does not necessarily mean you have to stop having fun or doing the things that you like to do. It doesn't mean you have to become a burnt-out, dead shell of a person, either. What it means is that you stop seeing yourself as the center of the Universe and that everyone must bow to your whims and preferences. It means that you should be more considerate of how your actions affect others. It means that you should be more willing to allow others the latitude to have the same freedoms of choice and expression that you enjoy (for example, calling someone an idiot or other pejoratives or lambasting them just because they use different electronic devices, play different games, or prefer a different gaming platform than you do). It's about recognizing your own mistakes and owning up to those mistakes rather than trying to blame someone else (i.e. personal responsibility and accountability). It's about learning to control your emotions, whims, and impulses rather than letting them control you; this does not mean you never act out of emotion, whim, or impulse, but that you are more decisive about when and why you do so (know yourself). It's about learning that you don't know everything and that sometimes mistakes happen; even further, to not get so emotional when the mistakes occur, but to deal with the mistake in a calm, reasoned and rational manner (player behavior in a MOBA, I think, is a good contrast of what I mean here). It's about being able to attain a balanced and stable perspective on life that is not so easily swayed by singular instances of happenstance. It's about being able to see yourself and others from a bigger perspective than simply what you want and what is best for only you. It's about learning to harmonize with others and being able to engage in a more stable, non-adversarial relationship with others. Growing up is about learning to live your life and letting others do the same.
Yes, you could say that by the above that there are a lot of adults, then, that need to "grow up", and you would not be at all wrong. However, just because you see some adults behaving poorly does not negate the truth or validity of the statement. It just means that they also have not "grown up".
Of course, here is one fact, no one can really make you do anything, not even grow up, if you truly don't want to; there is only one thing we absolutely have to do, and that is die (in the future, it may be possible that even that is not true). Just realize there are consequences, depending on the circumstances, for your actions, because no force of will (unless you are God) can change how the Universe works; it is up to you to choose the particular outcome you desire. If you are going to have the attitude that you get to do whatever you want, when you want, to whomever you want, for whatever reason you want, and no one better say anything to you about it, don't be surprised or complain when others visit that same ideal on you or start treating you poorly in retaliation for the poor treatment you give to them.
Ultimately, you are the one that decides your life and how you live it. However, this means that almost everything (pretty much 99.99%, barring natural disasters and other "acts of God") that happens to you is entirely YOUR OWN FAULT and the direct result of YOUR OWN CHOICES. So, you have to know the reason for your choices and understand your own mind well enough to know your true motivations behind your actions. There are no guarantees about anything in life; the Universe owes you nothing, not even the chance to live (e.g. still-born babies). Growing up is about learning to live with this and still not go insane in the process. Living your life to the fullest despite all the negativity that goes on in the world, despite all the failures and disappointments, and allowing others the latitude to do the same.
Notice, I've said nothing about having to stop playing games, watching cartoons (hell, I still do those myself, and I'm going on 42), or playing with toys. Nothing about growing up means you have to lose that child-like sense of wonder at the world around you. It simply means you approach life and others differently than you did as a child, particularly, you are no longer so self-centered, self-absorbed, and ego-centric. The old advice that comes to mind is having an old mind but young heart.
Apologies for the wall-of-text.