[Smirnoff]

The literal translation of the song is complicated and will not be clear to you... I will briefly explain the meaning and main message of the song:
A girl (singer) has been dating her beloved young man for a long time. He courts her beautifully and gives her expensive gifts. The girl is ready for a serious relationship and proposes marriage to the guy.

About the title of the song.
During intra-family (love) relationships, a man and a woman often informally call each other in a diminutive form with various words: fish, sun, bunny, etc.
A similar phrase is used in the title of this song, i.e. "Don't call me bunny", and instead of beautiful words, it would be better to propose to me... (It's hard for me to explain )

About the video.
There is a program on Russian TV called "Let's Get Married". Single men and women come there, talk about themselves and try to find their love. The video uses a humorous parody of this program.