To understand what is going on here it helps to undo the ellipsis. Ellipsis is the leaving out of one or more words in the second of two parallel constructions, in the faith that a reader or hearer can and will mentally supply them on the basis of the first.
- I have the same color eyes as my father [has].
- My personality is very similar to my father’s [personality].
The second pair is a little trickier.
- She was born on the same day as
me I [was born].
- Her ideas are quite similar to
mine my [ideas].
For #1 of this second pair, the substitution of me for I (only when the was is left out) is characteristic of colloquial and/or informal usage. Some will prefer and use I there even in the elided form; inverse snobs will despise them for it. For #2, the possessive pronoun my characteristically changes to mine when the word for what is possessed is elided, so when we restore that it reverts to my.
(In earlier versions of English, my also became mine when followed by a vowel sound, much as a becomes an; Julia Ward Howe was being retro in writing “Mine eyes have seen . . . .”)