"I have my own cats."
Translation:Jeg har mine egne katte.
"Mine" is used with plurals, regardless of gender, so must be used here with "katte". If it was just one cat then it would be "Jeg har min egen kat".
"Mit" is used with singular, neuter gender nouns, so it would be "Jeg har mit eget kæledyr" for "I have my own pet" but "jeg har mine egne kæledyr" for "I have my own pets". (Maybe a bad example because there is no change in "dyr*" between singular and plural).