Husse could be translated as 'owner', 'master' or even 'dog dad' — it's a word you use to talk about the owner of a pet.
It comes from the word husbonde, which as you might guess is related to English 'husband'. It literally meant something like 'dweller of the house' (bonde is related to the verb bo, 'to reside') and was used to mean husband, or head of the household.
Husse is only used for men, and the female equivalent is matte.
Matte is a shortening of matmor (literally 'food mother') which can mean either 'housewife', 'homemaker' or 'pet owner'. In other contexts, matte is a shortening of matematik, meaning 'maths'.
These days the words husbonde and matmor are just as old-fashioned as the traditional gender roles they describe, so you are unlikely to hear them in Swedish. But you will hear pet-owners refer to themselves as husse and matte.
There are other synonyms you could use instead if you prefer. If you're in a more formal setting, you should stick to saying something like 'jag har husdjur' (I have pets) or 'jag är hundägare' (I am a dog-owner) instead.
For more colloquial contexts, a word of caution about directly translating the English terms 'dog mum/cat mum'. Hundmamma and kattmamma in Swedish generally refer to the animal themselves, in other words to a dog or a cat with babies.
But you will hear Swedish pet-owners, especially younger ones, calling themselves mamma or pappa (mum or dad) instead of husse or matte when talking to their animal, for example 'mamma/pappa älskar dig!' (mum/dad loves you!) instead of 'matte/husse älskar dig!' (your owner loves you).
Jag vill vara en rolig matte
I want to be a fun pet-mum/pet-owner
Hunden gick fram till sin husse och slickade honom på handen
The dog went up to his/her owner and licked his hand