You do generally develop things like fine motor skills and critical reasoning skills based on age (or at least corollated strongly enough with age that you can you it as a reasonable proxy).

A child who is old enough to have learned "grasp the door handle to exit the building" and "food is in the cabinet" and "I can push some buttons on the phone to summon help" and "i'm supposed to poop in the toilet" is going to have more agency over their surroundings than a dog that pretty much just knows "I should poop outside if I can" and "this smells like food" and "sometimes if I bark loud people will come" in the situation I described.

The perception absolutely is a bad heuristic, but isn't wrong across the board. That's important to acknowledge because it means people are more likely to actually believe it and more likely to disbelieve you if you say "oh but that's 100% wrong all the time". Especially because the cases where it isn't true are more complex. For example, animals in poor households are perceived to have less control over being subject to poverty than the people who live there, when, as we know, poor people (adults and children) also have virtually no control over their economic circumstances.