The definitive answer, from an English language corpus is very clear: your friend is wrong. like me. is at least 1000 times more prevalent than like I.
So where did your friend get his crazy ideas from?
Very roughly speaking, the rule in "natural" English is that you only use I before a verb. In other situations, revert to me.
It's me! I'm over here!
He is taller than me.
He is taller than I am.
One day, me and my dog went for a walk.
I went for a walk with my dog.
In formal contexts, the influence of 18th century Latin-influenced grammarians has crept in, and there are still authorities who advise people to write like this:
It is I!
(never It's I, because It's would be sloppy and vulgar)
He is taller than I.
(many people will tell you that this is the correct because taller than I is an abbreviation for taller than I am)
My dog and I went for a walk.
(never I and my dog, because it would be rude to put yourself before your dog, apparently)
You can probably tell that I don't think much of this version of English grammar. But there you have it.
So it may be that one of these Latin-freaks has advised your friend to use like I. Or maybe your friend has taken the justification for writing He is taller than I., and applied it to He is tall, like I. Well ... at least he's trying to apply the rules consistently.