It is not neccessary that you add 'pet' in front of dog to specify 'pet dog', it is more a writing preference. The only corrections needed to make it grammatically correct are:
(1) When writing, all sentences begin by capitalizing the first letter in the first word in the sentence. (In this sentence tina should be corrected to Tina.
(2) All sentences end with some form of punctuation. This sentence is an example of a comparative sentence therefore, it should end with a period ( . ).
Corrected sentence shoul look like this:
Tina wants a dog for her birthday.
It's grammatically correct. But I think you mean a pet dog, to make it figurative you can use the word 'pet dog'. And the first word, Tina is a name so the first letter should be in capital, then it would be error- free.