But of course.
Part of it is language development... Kids read minute facial and body expressions more/better than they listen to language.
When very little (or scared and older) kids pattern off us. If you cry, your toddler will, too. If you're a ball of tension / snapping, you get a snippy stressed out toddler. When you're scared/nervous, they're terrified.
Cool tests show adults pattern the same way in countries they don't speak the language to a degree.
OLDER kids (and adults, when they speak the language) don't pattern... They REACT.
It's a neurological milestone... The switchover. And also the basis of 'marital conversations' that LOOK innocent to start by an outsider... But are predicated on long history of observation. In longterm couples studies, there tends to be a 'warmth' / "that's what they do" attitude, in unhealthy/dying relationships it tend to be an 'Aaaah! They're doing it again!" reaction.
Back to kids.
When we're stressed we're either stressing them out, too.. So they respond in kind / our own medicine... OR they're trying to STOP our stress. Either by getting hyper/happy/trying to distract us (just like a mom does with a toddler about to lose it in public) OR by 'disciplining' us, the same way we do with them. If you use short clipped speech when you're warning them to shape up... Guess what? They'll use different words (I want a Popsicle, my socks don't match, Joe's breathing again), but the SAME tone and speech pattern you use 'Counting to 3 / Full Name Here / Don't make me stop this car).
It's the cool blend. Kids are trying to interact with their environment. Using subtleties we don't usually notice... Because they're responding emotionally/copying tone, pitch, stance, etc., when we'd respond verbally. They're REACTING but not patterning... And don't have the cognitive development yet to use adult lexicon.