Interesting idea. But it's "I am", not "I have". So… I don't know…
There are some words that can be easily be used with normal nouns, but require the possessive forms when used with subject pronouns:
Araba gibi - like a car, but Senin gibi - like you.
The suffix "-(y)le" (with) is the same:
Arabayla - with the car, but Seninle - with you.
It's just a special rule.
Thanks! I'll just remember that those pronouns sometimes take endings where regular nouns don't, just like in English. In English, pronouns are the only words nowadays that change endings depending on their case (other than possessive). To the house, from the house, of the house, the house. To him, from him, of his, he.
Ektoraskan, look at the explanations in the TIPS. It is said that when "gibi" is used with a pronoun , here "you"="sen", the pronoun is marked with the genitive suffix, so "senin". https://www.duolingo.com/skill/tr/Prepositions/tips-and-notes