1. Anh ơi, mưa! Nhớ anh nhiều nhiều.
Honey, it's raining. Miss you so much.
2. Anh ơi, Mưa nhớ anh nhiều nhiều.
Honey, Mưa misses you so much. (In this case, the speaker's name is Mưa which means rain.)
"Anh" is a personal pronoun and object used for male. It depends on the context that we can translate it as "you" or "I". When the speaker is female, "anh" refers to "you". In contrast, if the speaker is male, "anh" refers to himself ( = I ).
For example:
Girl: Em nhớ anh nhiều nhiều. (I miss you so much.)
Boy: Anh cũng nhớ em nhiều lắm. (I miss you so much, too.)
Girl: Anh đang làm gì đấy? (What are you doing?)
Boy: Anh đang làm việc. (I'm working.)