there is no special verb in Russian , here is could be just присмотри за кошкой , собакой . it not the verb for just pets care . You could say - Присмотри за чемоданом - take care about the safety of my luggage

Взять на передержку - значит заботиться о животном во время отсутствия хозяев. Оставить на передержку - передать своё животное другому человеку на время отъезда.

I would expect to hear something like this form my friends: Specific requests: "привет, мы уезжаем на море, не мог бы ты покормить кота в моё отсутсвие? Ему желательно обновлять воду раз в два дня." "можно мой .... поживёт у тебя две недели?" Unspecific requests: My friends are mostly specific, but then I expect "присмотреть за": 'watch'. We apply "присматривать за" to things, pets, playing children... 'Позаботиться о' take care of is less likely. It may appear when I already know the implications... like, when I have already done this before for this person. Otherwise it may sound too serious and cause confusion... The owner wouldn't want me to suspect (even for a moment) that I need to take a serious responsibility! If I do it myself, I will describe my specific task or use "присматривал".

surely -Взять на передержку -does exist , but it is a kind of official expression , f.i. some wild animal is weak or wounded and some one берёт животное на передержку . i am not sure if he get money for that care from any state organization , perhaps yes , but i would never use that word asking my neighbor take care about my cat , perhaps somebody does , perhaps somebody's neighbor would has hard time to get what he is being asked for . Cat knows that .

And defintely for a person who doesn't remember/use this word it _sounds_ formal. Nothing light-herated about the way it sounds.