Relative Clause Ambiguous

After looking through a few questions on here, I came across the idea of the following being ambiguous.

私が好きな人

Apparently this can have the double meaning of 'the person I like' and 'the person who likes me'. Grammatically, I don't understand how this work. I was under the impression that が indicated the subject. Surely then the only interpretation is 'the person who likes me'?

As an additional question, if the above is allowed to have both meanings, then why can't 私は猫がすきです。Mean 'I like cats' and 'A cat likes me'?

So in summary, how do these meanings work grammatically?

Thanks in advance.