I'm pretty sure the ML is not telling about the name because he wants to see if MC will fall for him without knowing that.
Basically he is doing it out of his own selfish insecurity, something like, if two namers meet, they're only together because they have each other's names and it's not true love.
We've seen him being very sceptical about the namer thing.
I suspect maybe he saw two namers cheating on each other with no-namers and it traumatized him into believing namer thing is a scam.
I'm pretty sure the ML is not telling about the name because he wants to see if MC will fall for him without knowing that.
Basically he is doing it out of his own selfish insecurity, something like, if two namers meet, they're only together because they have each other's names and it's not true love.
We've seen him being very sceptical about the namer thing.
I suspect maybe he saw two namers cheating on each other with no-namers and it traumatized him into believing namer thing is a scam.
Annoying prick.