When is your birthday?/ What is your date of birth?

How to answers these questions correctly?
1. When is your birthday?
It's on June eighth.
It's June eighth.
2. What is your date of birth?
It's on June eighth.
It's June eighth.
Are both the answers for each question correct?

They are all OK, but "June eighth" would be fine for both.

Not A Teacher

To me 2. requires the addition of the year of your birth

That is often information a person does not want to share.

Those are possible, but without further context, not required.

I think the date of birth is not complete without the year.
Anyway, people do not ask about one's date of birth unless it is for official purposes.

Thank you all.
So the preposition "on" used in the answer to these questions is optional? (It's on June eighth.)

I think the answer to that will depend on whether you want, or have, to use AmE or BrE.

AmE: It's (on) June eighth. (Mike, or another AmE speaker, can correct me if that's wrong).

BrE: It's on June the eighth. (British English is quite strict about using this set form. As Piscean stated that could also be used as "It's on the eighth of June.").

Is it OK to say "My birthday is on June the eighth/on the eighth of June"?

To me that is the standard BrE way of expressing it.

For clarity: "My birthday is on the eighth of June" or " My birthday is on June the eighth".

In BrE you have to use "on", "the" and "of", as above.

I can confirm that Americans happily say "May 23rd" without the "the."