"Er ist genauso alt wie ich."
"He is (exactly) as old as me" should be an accepted translation.
I'm not sure if using "I'm" in the current correct translation "He is exactly as old as I'm" is technically correct, but I have never heard or seen anyone use that in 20 years, so I assume not, and even if it was, it would be bad to teach new english speakers to say that.
I think this is an exception due to the comparison: Er ist so alt wie ich. Das ist so breit wie hoch. Er ist so nett wie zuvorkommend. Du bist besser als er. That, what's being compared, all seems to be in nominative, infinitive or at least uninflected (I can't tell it apart).