KENNEL can be a cage in your house, or a building you take your pet to. We use IN a kennel for the first one, and AT a kennel for the second one.

But, Uly , If I'm not mistaken "передержка" is about a situation when people leave their homes and don't have opportunities to take their pets with them and they have to find a place to temporally keep the dog or the cat.

That's true, Маргарита! Uly, whenever I go on vacation, I have to leave my cat anywhere not very far from my home. Once my pet stayed with a girl who earned money that way, in her apartment. Could we call this a kennel? More over, in mass media you can come across ads like this: "Предлагаю услугу предержки животных" оr " Требуется передержка собаки". So, передержка can mean a place or service. I'm wondering...

@"Once my pet stayed with a girl who earned money that way, in her apartment. Could we call this a kennel? "
It was an "(in-)home pet boarding"
Another option is a pet sitter (coming to your home, can be your friend)

I think, "pet boarding" is the best translation of "передержка".
Added.

A kennel is a place of business, not someone's home. As grumbler said, that would be boarding your cat at someone's house or with someone.

Thank you, Tatiana, grumbler, Uly!

Thank you all!

👍🏼