Maybe 'unicorn' wasn't the correct word, how many of the products you absolutely love feature a character in their logo? Yes, it does depend on the context but you can consider the context to be as appropriate as octocat is to github.
>In my opinion having the character in the logo is crucial to the marketing. How do I persuade her that we must do this?
These statements seem to conflict with each other.
Taste is highly subjective, but your designer is a specialist in visual communication and literally sees the world in a different way from how you do.