猫と暮らしていますって英語でなんて言うの？

When talking in general, we tend to say, "I have a cat or a dog", Our pets live with us, just like people do.

We can actually use both: "I live with my cat" or "I have a cat".
We do the same when talking about our family too.
"I have two kids" and normally our kids live with us, so it doesn't mean ownership, when we say, "I have".
"Have" is an auxiliary verb

"My cat lives with me!"
"My cat resides with me!"

"Resides" means that you have made a home with your cat

"I live with my cat." this explains that you share your house with your cat.

"My housemate is a cat." This explains that your cat rules your house as much as a person would.

>My cat lives with me.
>My cat and I live together.

• By saying that you and the cat lives together indicates that you have a cat as a pet that lives in the house with you.
  Both sentences clearly indicate that your cat lives with you.

Well, I hope the poor cat belongs to someone! There are stray cats of course that don't have a fixed home and wander from place to place looking for food and possibly attention - I suppose this cat may be a semi-permanent resident or sometimes visitor to your home.
Phrase one tells the listener about your living companion that is not yours.
Phrase two is just a comment about a cat that stays at your home!

When explaining you have a cat, the following expressions would be useful:

I have a cat.
I have a cat as a pet.
I own a cat.
My pet is a cat.
I am a cat owner.

You can say :

I live with my cat.
My cat lives with me.
My cat and I are living together.
It's just my cat and I that lives in this house.

examples
"I like cats, I have two pet cats."
or
"I have a pet cat."
or
"I have cats."