"dört tane kuşum" means "my four birds" (or "dört tane kuşum var"= I have four birds)
but the sentence above does not indicate any possession so there is no suffix on "köpek"
tane is optional, you don't have to use it. but it is very common to use when you indicate the number of something. You cannot use it when you talk about age, or quantities with units (iki kg, üç metre, dört dolar, beş lira, etc)