I mean the etymological/ancient/root/deep meaning or a meaning from an era like 15th century
Well, we should definitely distinguish "etymological" from "deep"; nothing about the past makes a meaning any more valid than a meaning from the present. (Hence, I'm keeping the bottom half of this answer for anyone wondering whether this à is of any importance to a learner.)
Going by the TLFi article and the Latin root, where jouer is from an intransitive verb meaning « s'amuser », I suppose it makes sense to speculate that Vulgar Latin speakers used it more as you suggest: « Il joue, il s'amuse. » — « À quoi ? » — « À ludus latrunculorum. »
My logic is that the French people would not have been using 'à' without a reason. Because in French language, there are many verbs used without a preposition.
The reasons for using or not using à aren't related to the meaning. There's no generalization I know of that would link verbs that take a prepositions vs. verbs that take no preposition. Again, in English we "listen to the radio" while the French « écouter la radio »; in French, you « regarder à la télé » while the English "watch TV" — yet, despite the inverted prepositions, these pairs describe one and the same action.
Also, according to English speakers to play is an activity that does not need a preposition, too.
It's tempting to explain one language by another, but these are unrelated words in very distantly related languages. What English does with "play" has no particular bearing on what French does with jouer.
Original answer regarding modern French
No, that's not the true meaning, for several reasons.
The type of complement a verb takes is a matter of grammar, not semantics. Otherwise, I take it you'd interpret « il joue du piano » as "He's having a good time with his playing of the piano"? It gets silly fast.
The meaning of a French sentence is unrelated to English translations. That is, if there were some other "true meaning", it would be expressed in French and it would actually be hard to express without relying on this same à (« il s'amuse [à jouer] au tennis »).
But in fact such a sentence doesn't capture the true meaning, as can be seen by the many contexts in which we'd say « il joue au tennis » when a person regularly plays tennis or is playing tennis at this moment — whether or not he's having a good time! In other words, the true meaning can be seen by how the sentence is used, and it's just used to mean "play".
Even if you were to translate in this literal way, à is not always translated at. It has no exact translation.
In short, this word-for-word reading is neither accurate nor does it help comprehension. It only hurts it. If you love grammar, you can take an interest in why jouer takes à, but if you simply want to understand and speak French, it's a fairly meaningless detail.
A French speaker might as well say: "Why in English does listen take to, as in He listens to the radio? Is the true meaning « Il apprend attentivement en ecoutant la radio » ?" The answer would be the same: no. "Listen to" is the natural, and only, way to describe how one interacts with a radio; and « jouer à » is the natural, and only, way to describe how one participates in tennis.
Also, as Lambie points out in the comments, this is a standard construction with jouer, so you'd have to propose a similar alternative meaning for a ton of phrases.