I think that your second paragraph presents a false dichotomy, and things go awry from there.
The classes and objects in a computer language do represent things and actions, if only via a metaphor involving the programmers. But computing is clearly an abstract rule-based game. So the 'game' model does not remove 'representation'. If the program's output has an intended use, both models are involved.
They are separable, though. Consider a field of math like Group Theory, there the representation can be moved from one field of reference to another at will, as the goal is to isolate the behavior that all potential references share. Or even back off to counting. Abstractly 2 + 2 = 4 is not representational, as there could be two of anything involved, or you could simply be rehearsing empty rules for a future potential employment with referents.
So there are languages that are not representative because the statements made are never bound to specific referents. (Even when the process of the argument establishes potentially important facts about many given, relevant sets of applicable referents.) So, in a narrow sense, we observe exceptions, and your answer is 'no'.
It is tempting to create a place where those things 'live' and invest in logical or mathematical Platonism, but that only adds unnecessary complication. It seems possible, if we go there, that we can completely separate the two, and have the concrete 'participate' in the abstract. But it fails to explain why only a very narrow range of complete abstractions appeal to humans and get involved in our languages.
If there is all that real abstract stuff out there, what is special about Groups or Numbers? Clearly they are special, at least to us. And (I would propose) that is because we intuitively feel we will find referents for them. So are the elements of this mythical domains actually referents to begin with, or are the statements just empty forms seeking referents, to which they will be bound later?
So Platonism nets you a 'yes' answer to your question. But artificial constructions that stretch to cover the data and raise more questions than they answer are bad science, and I would put that 'yes' aside.
We are left with the observation that language may be nonreferential, but it loses meaning if it is truly free of referents, at least in the sense that we intend for abstract reasoning to have future concrete referents.
At the other extreme language is never truly bound to its referents, either: some definition involved always involves abstracting away elements of the concrete -- 'my dog' does not contain exactly the same set of molecules at the start and the end of any given statement about her. She is a furry abstraction.
It is more reasonable to see abstract formalism and real semantics as the opposite ends of a semiotic spectrum between which we interpolate in use according to the specificity of the ideas involved.
Language is an interlocking set of games of varying degrees of referentiality. Different important points along this continuum play different roles in communication and reasoning. Near one end, powerful languages like higher math are non-referential. Near the other deeply referential languages like literary descriptions give no leverage for reasoning but are useful in their own right.