why is correct MY grandmother needs a smaller dog? and not the grandmother needs a smaller dog? the grandmother can be yours
In English, in this sentence, "grandma" or "my grandma" would be used quite often.
In contrast, "the grandma" and even "a grandma" would be technically correct grammatically, but they would be used only in special situations.
For example, you could say "A grandma needs a smaller dog" if you wanted to say that in general, older women need to get small dogs rather than big ones.
Or suppose someone pointed to three women at a pet store and said, "Here are a grandmother, a mother, and a daughter. Which one needs a smaller than average dog?" You might reply, "The grandma needs a smaller dog."
But except for special situations like the above, "the grandma" would not often be used.