Yep, those are the rules
Now the rules for carrying things aren't really made to make sense, they're made to build good stories focused around mostly Medium sized people doing heroic stuff. So a lot of the rules aren't really focussed on realism.
Pretty much everyone in the D&D world has a level strength that would be unreasonable in the real world, and it gets worse for smaller creatures because of the flat modifier.
For example, by the normal rules, an average strength human can lift a weight of a whopping 300 pounds. (10 × 15 × 2 for lifting a heavy load). I happen to know that one of my colleagues in office who is a weight lifter complains about not being able to do that. He is definitely no weakling, but would score no more than an 8 Str in a D&D world. I, myself, not being into heavy lifting, would be lucky to have a 4 Str, even though I look like an average (but large) human in build.
On the other hand, a D&D human with excellent (20) Strength would be able to lift as much as 600 pounds. That's beyond modern world records. And these aren't dedicated weight lifters, they're just strong heroes and breaking every world record in the book is just another thing they do in addition to beating up dragons.
So yeah, your friend is absolutely right that in the D&D world an average human can lift things that would make your jaw drop, and it scales down so that all creatures can (by the rules) lift what sounds like unreasonable amounts. Including, if you just follow the rules, rats.
So yeah he's right, and if he wants to play that way, a rat can lift 15lbs.
Must be a big rat.