Listen, I know that in Hello Puppets they most likely refer to each other like family in a cult context but I cannot help but imagine that Riley and Nicks rivalry in the show is similar to a sibling dynamic in the way that it's petty and goofy! So this is probably more so of a headcanon on their dynamic if that makes any sense (since we don't get to see it in action merely hinted at in various ways).

I'm talking like they argue about stupid pointless stuff, whose better at what. But! Sometimes they burst into each other's room unannounced just to wack something off a nearby table or desk, stare, say nothing and leave. And obviously they don't close the door on the way out to be more obnoxious. Or worse, be the most obnoxious person existence and start asking really dumb stuff or just start screaming unprovoked and like not in the way where they're blaming each other for something (which they've probably done) I'm talking just full on screeching.

Or the classic pointing and saying "your gay/a bitch/ugly/etc" unprovoked just to be an ass.

Continuing on the leaving the door open I feel like Nick specifically has slowly opened the door long enough for it to make a long and loud ass squeaky screeching noise while Riley was trying to work. Nick most likely got chased out.

Daisy has most likely had to grip them by the ears and lecture them on being nicer to one another.

As for Mortimer I feel like he doesn't do too much about it, but in canon he most definitely has yelled at them to knock it off atleast once.