Hi,
I wish I could speak English as fast as I speak Japanese.
I wish I could speak English as fast as I can speak Japanese.
Are they correct? Yes.
Can can be omitted? Yes.
Is there any other word can be omitted? You could say it this way.
I wish I could speak English as fast as I can Japanese. Some people might put a comma before 'Japanese'.
I wish I could speak English as fast as Japanese. However, as you start to leave a lot of words out, you have to be careful that other interpretations are not possible. The context will normally makes the meaning clear.
Best wishes, Clive