"Mne" a "mě" should indeed be both accepted, but it is not true that the pronounciation is the same. "Mě" is pronounced with soft "e", but "mne" is not.
The only difference here is that "mne" sounds a little archaic and is almost not used in spoken Czech anymore.
But also don't rely on the theory "sounds the same = can be used as well". "Mě" and "mně" sound the same but are used in different cases so it actually never happens that both of them are correct, always just one. But even Czech people struggle with that and a lot of the times just don't know which is correct, so don't worry about that :-).
For me, personally, using pamatovat with a direct accusative, without a reflexive pronoun, mainly means to be old enough to remember something.
Doptej se, zda mají v rodině někoho, kdo pamatuje válku nebo sametovou revoluci.
Most v Kuněticích pamatuje válku a bude sloužit dál.
Naštvalo mě tahání, nemocnice pamatuje válku.
Armádní vozidlo, které pamatuje válku, je třeba hlavně pořád promazávat.
But dictionaries also show this usage for normal remembering something and it is accepted in this course. I would prefer "pamatovat si + accusative" or "pamatovat se na + accusative".