"the same as me" versus "the same with me"

(Grammar Questions I Can't Answer)

In speaking activities, I find myself constantly correcting my Vietnamese students when they say, "The same with me."

"No, it's the same as me," I'm always saying.

One day, a student asked me, "What's the difference between the same as me and the same with me?"

"The same as me is correct," I answered.  "The same with me is wrong."

But the students had actually looked up this question themselves, and there were several grammar forums online discussing the difference.
One example HERE

"The same as" - comparison. Two or more things are the same.
"the same with" - Could say, "It's the same with me". In agreement with something

...which, yeah, come to think of it, that's right, isn't it?