[…] Bob has left me with the impression that he believes he is my manager. Saying things like, "When are you going to get product X to me?" or "Is there anything important C-Level Boss should be made aware of?" This suspicion was confirmed tonight, when [he] said something to my coworker to the effect of, "Shouldn't I as their boss know when they are running a training?"
There are basically three possibilities here:
There has been a misunderstanding, and Bob genuinely believes that he's supposed to be your boss (and that you're simply undermining his authority by refusing to keep him in the loop and ignoring his instructions).
Bob knows he's not officially your boss, but your organizational hierarchy is vague enough (or Bob is delusional enough) for Bob to believe that by acting like he's your boss (but without actually claiming that directly in front of anyone who might contradict him) he can actually become your boss, at least in fact if not in name.
The misunderstanding is on your (and your anonymous coworker's) side: Bob isn't actually trying to act as your boss, but his department simply has legitimate needs from yours and he believes it's appropriate for him to contact you directly about them instead of going through your supervisor. (I'm including this possibility mostly for completeness, since others have suggested it.)
Your own assumption seems to be closest to option 2 above, and indeed the direct remarks from Bob to you that you've quoted support it (or perhaps option 3). However, if your coworker has reported Bob's words to them correctly, that actually makes me lean a bit more towards option 1 — it would be very bold for Bob to outright say he's your boss, even to a third party, if he knows he's not (and hasn't yet established a clear pattern of acting as your de facto boss). The risk of those words getting to you (as indeed seems to have happened) and backfiring is just too high.
Ultimately, the way to resolve this situation will depend on which of these is actually the case. The problem, of course, is that you don't have enough information yet to determine this for certain. As such, you'll need an initial approach that will get you closer to finding out what's actually going on without too much risk of bad consequences in any of these cases.
Going to your actual supervisor and getting them to confirm that Bob indeed isn't your boss was a good first step, in that it let you rule out option 4 (of Bob actually being made your boss without you realizing it). It doesn't seem to have led to any further progress, since your supervisor apparently isn't willing to discuss the issue with Bob on your behalf (or at least they haven't promised you that they would do so).
So, how should you proceed?
The following is at least to me quite clear:
My actual boss is not going to clarify management roles with Bob, as they are essentially equals
Bob will continue to treat me as his subordinate as it is his assumption that I am
It is not my role to tell Bob that he isn't by boss.
Questions: Are my assumptions faulty? How should I proceed? I value my relationship with Bob but also find his imperious inquiries taxing.
Under these circumstances, I would question your assumption 3 and suggest that, since your supervisor hasn't indicated their willingness to do so, it might in fact be time for you to have a chat with Bob about this yourself.
Obviously, it matters a lot how you do this. Confronting Bob in the middle of a meeting and yelling "You're not my boss, I don't take orders from you!" is… probably not the optimal approach, either for getting the issue solved with minimum drama or for preserving your friendly relationship with Bob.
(Even that might be better than just seething and doing nothing, though.)
What I'd actually suggest, however, is that the next time Bob says or does something that makes you feel like he's trying to boss you around, you say you have some concerns and ask him if he'd have time for a brief one-on-one discussion after the meeting (or whatever you're currently doing).
Then find a private place to have that one-on-one discussion and explain the situation (and your feelings about it) to Bob, starting with the assumption that it's probably all been a misunderstanding of some kind:
You could start simply by laying your cards on the table and telling Bob that you've lately felt as if he's sometimes making demands from you like he was your supervisor, even though (as far as you know) he isn't.
Make sure to mention that you've already confirmed with your actual supervisor that Bob isn't your boss and that you have no direct reporting responsibilities to him (assuming you did confirm that, too).
This both deflects the interpersonal conflict (as Bob now has to take it up with your supervisor if they disagree), demonstrates that you have someone else on your side and it's not just your word against Bob's, and crucially clears the air just in case your supervisor in fact did already have a chat with Bob about this (or might do so at some later point) even though they didn't tell you that they would.
Ask Bob what he believes your respective roles and reporting responsibilities are. Be prepared for this to be (slightly or completely) different from what you think is the case.
If the differences are major, don't try to argue the matter directly on the spot, since that again just puts your personal authority against Bob's. Instead, just iterate your own perception of the situation (to make sure you both understand where you disagree) and suggest that you should seek to figure out together what your actual designated roles and responsibilities are. In particular, if you believe that there's a factual disagreement between you and Bob about the organizational hierarchy that could be answered by consulting an official org chart or asking someone higher up than both of you, by all means do so.
However, try not to play the "let's ask someone higher up" card on minor issues that the people higher up may have left open on purpose for you and Bob to sort out between the two of you. Not only does that risk making the people you ask feel like you're wasting their time with trivial arguments, but the answer you get back might be just a rubberstamp of whatever Bob (or you) suggests the answer should be. (In the worst case, that could actually end up making Bob officially your boss!) Or, if you really want to take that risk, at least try to sound like the more reasonable and solution-focused party and offer a clear and reasonable sounding way forward for senior management to approve.
Make it clear (if that's indeed the case) that if Bob and his department need anything from you, you'd prefer that they go through your supervisor. Be prepared for some pushback, though, as Bob may legitimately see this as adding unnecessary delays and bureaucracy.
As a compromise, you can suggest letting Bob request some things directly from you (if you feel that's reasonable) as long as he also keeps your supervisor in the loop e.g. by CC-ing the requests to him. (If he promises to do so but forgets, do remember to forward the requests to your supervisor yourself!) Another option could be to suggest a three-way meeting with you, Bob and your supervisor to explicitly define the scope of what Bob is allowed to request directly from you and how the extra workload from such requests should be tracked.
If there's indeed been an actual misunderstanding on Bob's part, this approach should hopefully go a long way towards clarifying it. As a side effect, it can also let Bob re-evaluate your perceived behavior towards him in a new light, hopefully clearing out any resentment that may have been building up on his side.
On the other hand, if Bob actually has been trying to "expand his domain" and take on a supervisory role over you that he's not supposed to have, clearly laying out your respective roles and reporting responsibilities (make sure to write these down and get your supervisor and any other relevant parties to sign off on them!) eliminates the organizational ambiguity and makes it that much hard for him to keep doing so. And of course it also lets him know that you're aware of what he's trying and don't agree with it.
The same also goes even if Bob has just been trying to cut through what he sees as unnecessary red tape by bypassing your supervisor and making his request directly to you. In that case, what you really need to do is agree on a workflow that balances both of your goals, getting any necessary tasks done promptly while keeping your supervisor informed of your workload and not overloading you with extra unplanned work. Having a discussion like I suggested above is probably at least a good start on that.
Finally, I should note that my advice above is (in part) based on your claim that you and Bob "still get along very well" and that you value your relationship with him. Not only does a friendly and respectful relationship between you make it more likely for such a one-on-one discussion to work out well, but conversely, friendly and honest communication is also an essential ingredient for preserving any relationship.
Had your relationship with Bob already turned sour, I'd be more likely to suggest involving a third party as a mediator from the start. You might still want to do that if the one-on-one discussion doesn't go well, but in your situation I'd keep that as plan B.