I have noticed one thing when watching opera singers perform in concerts. Most of them don't accompany themselves when singing. Here I am talking about piano and vocals. I am not sure I have seen anyone do this, ie accompanying themsleves when singing. All I could fids is Richard Tauber doing it in a video:Why is this? Is it really that difficult to play the piano when you sing?
It's really not so much the difficulty, it's the expectation of the genre.
Opera singers sing, pianists play.
Pop & rock musicians frequently do both at once.
The list of those more than reasonably accomplished at this is colossal - start with just Paul McCartney, Elton John & Stevie Wonder & go on from there...
It's quite possible to sing and play at the same time but there are sound physiological reasons why singing in an operatic style while sitting at the piano isn't a great idea. Good singing technique, particularly when projecting or singing at volume, has at its foundation an open, upright posture. Sitting down hunched over a keyboard with your hands together pronated over the keys is almost the exact opposite of how classical singers are taught to hold their bodies.
Pop/rock musicians often sing while playing keys, but they are not generally singing in an operatic style or at anything like the same volume. If you're singing into a microphone you can get away with a lot of poor technique which will really show up if the power goes down and you suddenly have to fill a concert hall on your own.
Tauber was definitely 'showbiz' by upbringing and by inclination! He was notorious for being able to do everyone else's job as well as his own. Apparently, during the London run of 'Old Chelsea' he would sometimes decide to conduct the pit orchestra - doubtless favouring the audience with frequent winning smiles over his shoulder as in the video!
An operatic singer normally prefers to stand and face fully forward towards their audience. But it's not hard to play and sing simultaneously (at least when it's not a virtuoso piano part - I wouldn't fancy trying it on 'Erlkönig') and arguably it can give the most responsive, musically integrated performance.
Singers have a reputation for only caring about voice and being musically illiterate. There's an atom of truth in this - you CAN become a fine singer without reading a note of music. But many are also accomplished musicians. Remember that Tauber WROTE the music for 'Old Chelsea', including his big hit 'My Heart and I'.
It's not hard to sing while playing an accompaniment, chords, bass, strum patterns, arpeggios.... It's a bit difficult is to sing while playing a counter melody or something rather complex. It's not too hard to sing and play from music at the same time. One can play the piano and converse if the music is there (seems as if one reads the music and plays using two brain channels and converse using another) but it's hard to play from memory and converse at the same time.
"Is it really that difficult to play the piano when you sing?"
I would answer that you could learn a skill by practicing it every day. I learned pop singing + pop piano at the same time when I was 19 years old, and it didn't take me very long. Maybe only a couple of months, because I already knew how to do both independently (and it is easier to learn things when you are younger, I think).
Nowadays I can, for example, play the piano in a band and sing harmonies to the lead singer at the same time. But if I have to concentrate on singing lead and accompanying myself at the same time, I usually make the accompaniment very simple and have to leave the "more fancy" playing out to succeed in the singing well. And that is simply because I don't practice them together often enough (I usually concentrate on practicing singing only, I want to practice opera singing specifically and in that piano accompaniment is provided by somebody else, like previous replies have indicated)