As you said for yourself, there is relevance to the present situation. So grammatically present perfect would come there.
Why did the person emailed me "I did not forget you" [...]?
Well this maybe because he is not a native speaker. Or he might have thought no-one would notice the mistake or the slip-up. Or he might have thought that this is the correct tense (even after being a native speaker). Anything could have happened. By the way, the correct sentence should be:
Why did the person email me "I did not forget you" [...]? (Simple Past for emailing instance)
When reporting it to someone else, you could use either of them¹ did not forget or had not forgotten, given the first person said it in simple past tense: "I did not forget you"
He had emailed me saying that he did not forget me/had not forgotten me. (Past Perfect for emailing instance)
However, if the first person wrote it like: "I have not forgotten you", then you must use had not forgotten. This is called backshifting the tense.
He had emailed me saying that he had not forgotten me. (Past Perfect for emailing instance)
¹ Here is one reference site where it is stated that simple past in reported speech could take either simple past or past perfect (refer to the table). But there is ambiguity in the context then. The third person would think the first person said it in simple present tense which you converted into simple past tense or simple past converted into simple past tense.
However, this is the only site I could find which follows such rules. Majority of the site say otherwise.
According to this and this site, simple past goes into past perfect, no exceptions. But then again, past perfect could mean the original sentence was either in present perfect or simple past.
Conclusion: This is particularly an ambiguous topic in itself. Although people would rarely notice any omission or slip while in a talk, it may matter in educational outcomes and written format.