What kind of relationship am i in?

He says he doesn't want a relationship, made it clear.

But he:
Texts me good morning, good night and we talk or hang out every single day.

We go on dates together. He'll take me to a nice restaurant or museum, he'll even hold my hand in public. We'll watch Netflix together and lay in bed cuddling, he'll tell me he's so lucky to have me.

He gets me flowers randomly, buys me other gifts.

He often tells me I'm the most beautiful woman he's ever seen.

When we went to a club recently and a man was hitting on me he didn't like it and told him to leave me alone. And when he is hit on by other women he actually says he's in a relationship.

He introduced me to his brothers and his parents and his family call me his girlfriend but he doesn't because he says he can't be in a relationship.

He's not gay, he's straight.

I don't know what this relationship is. What's it called?