The most common and most likely answer to this crossword clue is the 6 letter word CANINE
I've seen this crossword clue in The Times
C
A
N
I
N
E
More clues leading to the result CANINE
- Dog carrying stick around at home
- Stick around at home, like a sort of pet?
- Dog beginning to cross a square
- Tooth circular initially, then a square
- Dog tooth
- Like dog’s home, stick around!
- Stick around at home for the dog
- Doggy’s stick to carry home
- Animal container, popular, English
- Hound men in action partly retreating
Buddy explains!

The straight part
We know that "like a sort of pet?" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case like a sort of pet? = canine.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does CANINE mean?
CANINE noun- Any member of Caninae, the only living subfamily of Canidae.
- Any of certain extant canids regarded as similar to the dog or wolf (including coyotes, jackals, etc.) but distinguished from the vulpines, which are regarded as fox-like.
- In heterodont mammals, the pointy tooth between the incisors and the premolars; a cuspid.
- A king and a nine as a starting hand in Texas hold 'em due to phonetic similarity.
- Of, or pertaining to, a dog or dogs.
- Dog-like.
- Of or pertaining to mammalian teeth which are cuspids or fangs.
- Of an appetite: depraved or inordinate; used to describe eating disorders.