Rude to ask about koseki tohon to dad?

Hey! Me and my dad haven't meet each other in person since i was born. We still have contact because he's supporting my education but sometimes I just want to ask him if he ever wants to acknowledge me legally. Tbh, i really want him to acknowledge me as his child. I am thankful that he supports me and stuffs but knowing that I am not his daughter in his registry really feels bitter in my mouth. I really dont want to make him think Im rude for asking such question but I can't help it. I feel bad for feeling something like this. Will it be rude for me or not?

I think you know your father better than anyone else.
Just find a right and suitable time to ask.

even after he acknowledge you, your nationality is not changed.
the difference between acknowledged or not-acknowledged is the you will have a legal right to inherit his heritage after he dies.
after the acknowledgement, you will have no change in your life in your country, but, his family in Japan will have a serious change.
you have a right to ask him that you are acknowledged. but, before to do that, you should ask his opinion how he wants to treat you in future.
generally speaking, English words are so straightforward and is not so polite and gentle. you should use the words carefully.
of course, it depends on case by case. especially when he is rich and he has a family or relatives, it is not easy to solve this problem peacefully.

By the way, what you're asking is "ninchi", not "koseki tohon" which is an official copy of what is already registered. It's not rude, of course. You have every right to ask. But it's a touchy and serious issue, and you need to be ready for a "no" as an answer.