I was also a son raised by a single mom.

My take: I certainly didn't have authority over my mom, but I definitely felt like I had a lot of responsibility. Households are typically run by (at least) two people, and for good reason. I stepped up where I could--helping manage the dog, helping make food, staying out of trouble when she couldn't watch me, in general trying to keep stress off of my mom--not because I felt like I was in a position of authority, but because there wasn't another adult to help shoulder the load, so some of the load fell to me.