Well I don't agree with Schopenhauer's theory, but I think he says to that that happiness is a very elusive and short-lived emotion, and if we ever achieve it, it is only to get frustrated by something afterwards. And, according to him and to what I understood, only suffering and obstacles, small frustrations grab our attention but happiness does not.
According to Schopenhauer, it's not some nerve or something that tells you you're happy (so in this sense and to finally answer your question, according to Schopenhauer still: no we can't physically feel when we are happy). Of course back then we didn't know about oxytocin, dopamine, and serotonin...
And obviously, Schopenhauer's theory really clashes with the epicurian theories "live the moment" "carpe diem" "happiness is the road, not the destination".
Personally, for many years I was only sensitive to the loss of happiness! Like, in the moment, I could not see it, but if I suddenly lost it, I could see in retrospect how happy I actually was. But on the moment I took it as normal, as granted.
It's only very recently that I was able to change this, and be aware that now is a happy moment. For me to be aware of it, it requires introspection, meditation, calm, boredom... and it's sadly not automatic.
I also would like to add that happiness in itself is meaningless. It's illustrated for example in Greg Egan's book "reasons to be cheerful" - you can be super happy, but if it's not linked to a reason (like, a significant person for example, or something you really want) it's all useless. I'd be interesting to have the opinion of a psychologist on that matter... do antidepressants make HAPPY or just less unhappy?