This.
That extreme/violent/vitriolic a reaction, combined with his work colleagues clearly not knowing he has a wife (who then sought her out to clarify who she is and reassure her she's not embarrassing etc. when they clarified she is indeed his wife).suggests intense, ground swallow me up level of embarrassment and shame for him; based on lying by omission about his wife (& child?) and being caught out in that.
He was caught out very suddenly and unexpectedly, he panicked anf became really angry etc. I would say the "fuck off" was the result of sheer panic and trying to keep op away from his table/self/work colleagues in the hope there would be no interaction that would reveal who she was. That obviously didn't work out for him.
The "you are embarassing" was a further panicked, angry attempt at getting her out of his vicinity where her identity and supporting facts were revealed and he'd be forced to acknowledge those facts.
Now the important question...... Why did he lie by omission to his work colleagues about op's existence (and his child's?) and why did being caught in that lie cause him to go into a state of sheer panic and anger??!!
Op, he's denied you ... And he was extremely panicky and angry when caught out in that.
I don't believe its just keeping work and home life separate. Lots of ppl keep those separate but they do not lie by omission about the existence of their spouses. It's the most natural thing in the world to mention them in passing when making small talk or speaking about practical arrangements.
That would not be in any way normal by even the most private, discrete work colleague.
The work home balance thing is total BS.
Balance means how much time and energy you expend in those two things. Absolutely nothing to do with the totally normal, natural acknowledgement of your spouse and any kids.