She doesn’t consider me a close friend but I consider her one/mine. She doesn’t see me as a close friend but I see her as one/mine. ‘consider’ and ‘see as’ are similar and can be mixed and matched, but I slightly prefer being consistent.

I'd say "like" is not quite right here - we don't say "like the way", we say "in the way". And your phrasing at the end is wordier than it needs to be, you can use this really brief and (I think) quite elegant phrasing: She doesn't see me as a close friend in the way that I do her. Having said that, why not replace "in the way that" simply with "as": She doesn't see me as a close friend as I do her. It looks a bit weird written down, but in speech it's good.

For 'consider' we would 'consider her' rather than 'she.' Rather than 'is,' 'as' sounds better, I think. You're considering her in the role/quality of 'as a close friend.' 'She doesn't see me as a close friend like the way I consider her as mine.' Or, including 'that' may be preferable to some: '...like the way that I consider her as mine.' You could also use 'to be:' '...like the way I consider her to be mine.' This is slightly more clear and maybe slightly less literary-sounding.