Or are we conditioned by e.g. school, TV/the media, other people?
You don't have to watch puppies, kittens, bearcubs, dolphin calves, otter cubs and primate young for long to recognise the games our children play. Tag/Chase-Me/Faster/Can't-Catch-Me and their variants, along with play-fighting (learning attack and defensive skills) are common amongst mammals, as you say. They are the basic survival skills to catch rapidly moving food while avoiding being eaten either by the food or the more rapacious members of your own kind - many species recycle their weak young by consuming them, and cannibalism has been evidenced in early European as well as in many tropical tribes. The same has been evidenced in dinosaurs and birds, and it seems likely that some dinosaurs did learn through play, but predatory birds learn to kill very quickly and any play stage is generally ephemeral - the behaviour is either hard-wired or swiftly copied form the parent. Knowing the basics of 'Staying Alive' from an early age would have given many individuals a big advantage, but being able to run and to know when to run and how to run effectively and to where (etc) would be pre-requisite to being entry-level players. Perhaps it is more a case of being hardwired to play the game of Learning, for which there is a variety of levels from passive observer (at birth), to novice (childhood), juvenile adept (various levels up through the teens), and so on. Human games become more sophisticated with age, as we can see in soap operas and dramas, but they can also shift their form - Staying Alive in mature adults often becomes one of Avoiding Early Death even in those who are convinced they will live forever after death ensues. If I were to speculate at a more subtle variance in human game-learning, You'll have noticed that many adults continue to play the games of their youth well into their maturity. This (and other details) leads me to speculate that the ability to learn HOW to learn varies quite widely, but any individual will still happily recognise their personal ability as quite normal, regardless of where they might actually be on the overall scale. So, extending your last point, some are very good at developing new rules and rule systems, even to the extent that millions of others find them intuitive (the game of iPod Operation, say), whereas others saw the Nazi game of 'Juden Raus' as a game, rather than conditioning, and the current Logo board game (UK/online) or the Car Logo game (US) may be viewed by current children in the same way, but then marketing people have always been remarkably good at playing with human wiring and our concepts of 'winning'!
I have some empathy with this view, as it is a rather chicken and egg question! Many would take the view that we are conditioned because we are generally penalised if we fail to 'play well with others' at an early age. Indeed, it could almost be viewed as a form of Stockholm syndrome, where the child is in effect a hostage to its parents and peers and the only way to escape is to play along, and the only way to convince your captors convincingly is to appear to be convinced yourself, at least until you can break free - but prolonged captivity can lead to your conviction becoming irreversible. Stockholm syndrome might be also be viewed as an extreme play in a game called 'Escape'. The question then becomes one of 'are we hard wired to become conditioned?' or one of 'is conditioning a game in itself?' - the game of 'Rewiring' perhaps. Given that there is a range of susceptibility to conditioning, presumably giving a Normal Distribution if you plotted the variances graphically, there's probably a genetic variation underlying the phenomenon - and different degrees in the ways a population perceives it.
You asked, 'are we hard wired to become conditioned?' Absolutely; no question about it - and nothing sinister about it, either. Your Stockholm analogy suggests that we are essentially free (unattached) individuals for whom any loyalty - even filial loyalty - is a form of undesirable captivity. But, in fact, we become individuals only through our relationships with our early caretakers, mostly moms. To put this into a framework closer to yours, it is not that we are hardwired to create games, but that we are created by games. However, the games that create us were in turn created by our forebears, ad infinitum. From this perspective, it is not that playing games is one thing we do, among others, but rather that the concept of games serves as an interpretive lens for understanding everything we do and are. But it, too, is only one such lens, among many.
Like many animals, we are clearly hardwired from birth to play.
Since games have rules (in fact, any game is defined by its rules), it would be easy to conclude that we learn (or "are conditioned") to play games; that is, to channel our play instinct into rule-bound forms.
On the other hand, I suspect (I do not know) that we are also hardwired to develop rules and rule systems. Since that is essentially what games are, it seems to me that we are, indeed, hardwired to play games.