No, there should not be an identification issue - unless rho=0. The reason is that e_A will introduce autocorrelation in A_obs, while A_me will not. This prevents the two shocks from being observationally equivalent.

Could I ask you another case?
If the measurement equation is

where x is a “scale” parameter to adjust standard deviation between observed data and shock A , and x should be estimated; A here is i.i.d shock with 0 persistence, so it is just the innovation. I also need to estimate standard deviation of A. Would there be identification problem in such case?

Yes, there would be a problem. You cannot estimated x and the standard deviation of A at the same time, because both capture exactly the same thing. You need to normalize one.