If it bothers YOU, then it is cheating. Doesn't matter if he thinks it is cheating.
His behavior indicates that he cares a lot about how HE feels, but does NOT care how YOU feel.
Don't stay with someone who doesn't value you.
And he lies to you and you buy in it, so you stay.
Time to break up. IF you want to be happy.
I doubt if you actually love him. Unless you two have been living together for 2-3 or more years, what you are feeling is attachment, but that is a selfish emotion that has nothing to do with love.
Love means accepting the other person as they are, and that requires that WE face our fear, insecurity and hurt and work with our own emotions ... rather than demanding that our partner changes so WE aren't uncomfortable.
Only those who are whole inside and skilled at dealing with their emotions are able to love.