Yes! You can use the same actual business entity/LLC for registering as a Publisher with both ASCAP and BMI, however your actual publisher affiliation names with each society have to be different.
For example, if I wanted to setup my publishing entity, Smulders Publishing LLC, I would name it as follows, since I cannot use Smulders Publishing LLC for both:
ASCAP - Smulders A Publishing LLC
BMI - Smulders B Publishing LLC