"My grandmother can't remember me because she has got amnesia."
Translation:Min bedstemor kan ikke huske mig, for hun har hukommelsestab.
I guess one can say that bedstemor - grandmother, but mormor - grandmother and farmor - grandmother. That is, the mapping back and forth isn't symmetric with these words and it's incorrect to say that grandmother means mormor or farmor in Danish — it means both.
You'll need more English words to accurately translate mormor (something like "my grandmother on my mother's side").
"Has got" in English does mean "has received", in this case it's synonymous with "has", though I think I will change the English to be clearer, because I think "have/has got" is more of a British (though not necessarily exclusively British) phrase than an international English phrase