Note Part 61.31(c):
(c) Aircraft category, class, and type ratings: Limitations on the
carriage of persons, or operating for compensation or hire. Unless a
person holds a category, class, and type rating (if a class and type
rating is required) that applies to the aircraft, that person may not
act as pilot in command of an aircraft that is carrying another
person, or is operated for compensation or hire. That person also may
not act as pilot in command of that aircraft for compensation or hire.
Since you will have the Category and Class rating, then you may carry another person.
Your old license says:
Private Pilot
Airplane Single-engine Land
Instrument Airplane
Your new license will say:
Commercial Pilot
Airplane Multi-engine Land
Instrument Airplane
Private Privileges
Airplane Single-engine Land
You may fly under part 91 ASEL, AMEL, and IFR
You may fly part 119 or 135 (or other commercial operations) AMEL and IFR1. You may NOT fly part 119 or 135 or any commercial operations ASEL. If you want to fly ASEL in a commercial operation, you will need to upgrade your ASEL by passing another commercial checkride in an ASEL.
This is the back of my certificate. I took the ASES checkride when I had a Private certificate.

1. Assuming you have the correct medical and any other checkouts you may need