Testing the drowning part is rather simple: just hold your breath. If you cannot suffocate, you will be able to hold your breath indefinitely, if you can suffocate you will pass out and automatically start breathing again with no ill side-effect beside a headache. (Pro-tip: do not try this at home! It's really uncomfortable if you are not immortal)
An easy test for the invincible part would be to get infected with the common cold. If the wizard spoke the truth, then the infection should be defeated without any effects other than a major hunger. As that illness is highly infectious, not getting it if you really try is close to impossible. Asking someone to cough in your face might however cause socially awkward situations.
Another test would be to find a sharp object and cut into the tip of a finger with it just deep enough to injure the skin. That part of the body is highly sensitive (it really hurts) and therefore measuring the time it takes to heal becomes very obvious as you get constantly reminded of the state of the injury. It usually takes about a week for such a skin penetration to heal completely, so if it is gone by the next day, that part seems to be true as well.
Testing for true immortality like driving a knife through the heart I would simply skip. There is no logical reason to test that. Sooner or later there will be a situation where someone will test that for me anyways. And if it still hurts like crazy there is no need to know, because I would want to avoid such situations byat all cost anyways. There is also the risk that I might pass out from the pain, and while my body would dissolve the knife I might not be able to eat/drink enough to support it, and then I'd die despite being otherwise immortal.
The hunger part should be a dead-giveaway alone, as a sudden overwhelming hunger would be quiet obvious. As my body approximately heals that way - sadly without the immortality and perfect recovery part - I got some experience with that, if I get ill I eat about twice as much on the day before (my friends keep calling me Frostmourne). Excessive drinking would only be necessary if your body actually needs to replenish fluids, but not for skin/bone/organ regeneration. Or if the body would try to flood out an infection, which probably won't happen with this type of immortality as such infections would be dissolved magically.
This form of immortality might result in the following side-effects:
- Excessive fear of starving, because you need food to survive, but can't store it, because the body dissolves it.
- Excessive fear of blade-like objects, because they could cut off your head.
- Allergy-like reactions when the body tries to dissolve otherwise harmless particles like spores or car emissions.
- Immunity to any form of drugs and/or medication, even those which could be helpful.