Two answers; one in response to your general query and one in response to your specific circumstances:
An emotional affair is when:
- The OW knows more about your primary relationship than your partner knows about the relationship with the OW
- There is secrecy
- There is sexual chemistry
Usually, an emotional affair is merely a prelude to a combined affair (physical and emotional). Infidelity doesn't have to include physical contact.
As for your personal circumstances, of course this is an affair and is infidelity. I would be astonished if nothing physical had happened, but it doesn't matter.
And you don't need to prove that an affair has happened, either emotional or physical. If it makes you feel unhappy and uncomfortable, that is enough.
I would however add that a solicitor is likely to agree with you that this is unreasonable behaviour, so if you were married, this would be grounds for divorce.
Don't mess around trying to get him to agree that he has done wrong - vote with your feet and how you feel about this is all that matters.