in my opinion, i think the right way should be "cats and dogs" but here is "cat or dog", why? is it right?

Well you can say either:My sister always takes pity on some stray cat or dogMy sister always takes pity on stray cats and dogs

only use the plural if you are saying "my sister takes pity on all stray cats or dogs' to say "some stray cat or dog" implies she finds one particular cat and one dog to pity and the word some isnt necessary.

My sister always pities on some stray cats and dogs.

It's right because you have the word "some". If "some" was missing then you could use the plural.

I think it is right. 'Some' means a non-particular cat or dog, single-form means it's just one.