It depends.
In the case that he repaired your car using a "rebuilt" axle - either from Audi or from an aftermarket supplier, then part of the deal is to return the bad "core" axle to the vendor. The rebuilder needs the old ones back to keep their awesome axle-rebuilding-business going. In that case, there will be a certain cost associated with keeping the old one, and not returning it as a core. That cost would be the "core fee". The penalty charge for not returning the core. For an axle this is usually around $0-$40. The price varies and depends on supply, demand, failure rate, the cost of new ones, and other factors.
In the case that a core fee applies to your replacement axle, AND you so choose to cover that fee yourself, then you can certainly have the old one back. It is, after all, your axle to begin with. Generally speaking, people don't ask for their junk parts back, so the mechanic will assume that you'd rather turn in the axle as a core instead of paying the penalty charge. He wouldn't even bother asking you.
BUT, if he fixed your car with a NEW axle, and no core charge applies - then he is obligated to return the old part to you if you want it. Unless, of course there was some prior agreement where he keeps all your junk.
That said, yes, I think you are smart to keep your OEM axle. Should your axle troubles turn into a revolving door of crappy, chinese replacement axles - you could either rebuild the OEM axle yourself, or send it in as a core to a high end axle builder like Raxles (who only take OEM axles as cores). Keeping the part is probably free, and leaves your options open. Now if the axle was clicking, it at least needs a joint. And if the core fee is $35, I might consider letting it go. You can get another junkyard OEM axle in good condition for that price.
All of that aside, some customers ALWAYS ask for their old parts back, regardless of their condition. This is their way of ensuring said parts actually got changed with new ones. And sometimes to satisfy their curiosity of how badly the parts were broken. It is usually best to ask before the work is done, as things will often get thrown away pretty quickly. Some people just want to see the parts, and then don't take them. But coming from the mechanic's perspective (note: I'm not a mechanic...sort of), it is not at all unreasonable for someone to ask about their old parts. Either to see them or take them home. In fact, most would probably be proud to show how bad the damage was, or how and why the part failed.
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