https://youtu.be/HuLhHpIdl88
a few months ago, my battery went bad and was replaced without much further issue.

A week ago, I had my alternator go bad on me, putting out voltage below the battery and causing the battery to drain. I took it into my local mechanic and they replaced the alternator for me. Immediately after I got the car back, I heard a decently loud whirring noise coming from the alternator area (see attached YouTube link) (https://youtu.be/HuLhHpIdl88). Does anyone know what this sound could be and should I be concerned about it? I tried calling the mechanic and he said something along the lines of "we checked everything before giving you the car, so I wouldn't worry" but that definitely doesn't sound like a proper answer to me.

 

That would be a good use for your stethoscope. Because that's not really a normal noise - Unfortunately on your V8 the alternator is not easy to get to. In fact it's REALLY difficult to get to. If your "local mechanic" has not replaced an alternator on a V8 Volvo before - he might not have been prepared for what it takes to do that job (and could have charged less in labor than what reality is!) - and might not be so receptive about doing it again it there's a problem with the part he installed. What part/brand/manufacturer was installed?

 

The part number is 36000791. Should be the OEM alternator for the S80/ XC90.
what has me semi-worried is in the parts/descriptions there isn't a new belt listed. From my experience with cars, you should almost always replace the belt when you are doing the alternator.
When I was talking with the mechanic it sounded like he knew what he was doing. He went in through the wheel well. I've gone to this mechanic before and haven't had much issue. Works on exclusively BMW, Volvo, and Toyota.