'Pentu' or 'poikanen' (cub, puppy or kitten) refers to an animal, so it would be very uncommon to use 'hän' [he/she]. Althoug I guess in English you could say "she has" for example of your pet or another animal mother, but in Finnish we'd say 'se' [it], and it's best to stick with the proper translations when teaching a language.
(Btw, you might hear some people referring to their pet as 'hän', as to elevate it above a "common animal", but it sounds kinda funny - at least to me.)
Yes, you can say a person has three puppies, as in owned or has possession of at any given point, but the 'it' here is your indicator that the sentence is referring to an animal. Finn's don't call people 'it', and they don't call animals 'he/she' (except for breeding purposes). So here, it is 'sillä on' (it has) instead of 'hänellä on' (he/she has).
¨The Use of the Adessive Case:
When something is ON something.
When something is NEAR something.
When talking about open places.
Possession (having something).
Using a tool.
With certain expressions of time.
In many phrases.¨