I live in a non-English-speaking country. A lot of people around me pronounce the letter N as "ain" (/eɪn/ in IPA). I am very confused because in dictionaries the letter N can be only pronounced as /en/.
Is this a mistake? Do any native speakers also pronounce the letter N as "ain"?
To answer your first question, the OED entry for the letter N (paywalled link) gives only /ɛn/ as the pronunciation in both American and British dialects alike. If you for whatever reason choose to view that dictionary’s pronunciation as prescription not description, then the letter N “should” be pronounced to rhyme with men and ten, not with main and feign.
That means that native speakers of English say the name of the letter N as /ɛn/ using the lax DRESS vowel, not /en/ using the tense FACE vowel the way you've described hearing it said in that non-English country you've so far only alluded to.
In some native-speaker accents these two contrasting phonemes do move around a bit compared with how they work out in other dialects, but they should still belong to distinct lexical sets no matter how they are pronounced. The same would hold true with words like gem with an open/lax vowel and name with a close/tense one.
To answer your second question, although I've heard native-speaker accents where pin and pen merge, I've never heard any native-speaker accents where pen and pain merge.
I have heard non-native-speaker accents where this happens, however...
I bet the people wherever you're talking about also pronounce the letter L to rhyme with nail not with bell the way native speakers do. Unlike English, some languages like Spanish do not distinguish lax vowels from tense ones. All five vowels there are considered close, and while some speakers may in some utterances say some words with a vowel that’s a little more open there due to its phonological environment, they cannot "hear" this as a separate vowel because they lack minimal pairs.
But because you can, you do, and this ends up being confusing.
That's why in a Spanish accent, the English word bit sounds the same as English word beat, and why English met in their accent would sound like mate to you. Maybe that's what you're hearing happen here: the original language doesn't distinguish an open/lax e from a close/tense one.
It turns out that name is a pretty good example for demonstrating variation in this phoneme across various English dialects. Follow that link to see those, and even hear them.
Notice how differently that works out in practice. Sometimes it’s one diphthong, sometimes it’s another, and at other times it’s not a diphthong at all.
Every single one of those is a “correct” but different pronunciation of that same word by native speakers from around the world. It's therefore “correct” in that accent but “incorrect” in others. What sounds normal in one dialect would necessarily sound abnormal in another that uses different rules for realizing its phonemes.
Phonemically those are all /e/, but phonetically they are certainly not always [e]! But this doesn't matter to our ear. Whether the vowel from name is ever some sort of diphthong there is a different matter altogether, one not especially important since that's merely a minor unconscious phonological effect, an offglide made by some speakers and not others. The contrasting feature distinguishing the two phonemes is whether the vowel is the open one as in DRESS or the close one as in FACE.
English has no minimal pairs differing only in whether there's a glide there in a falling diphthong versus a monophthong without a glide. So we have no minimal pair that has /eɪ/ in one word and /e/ in the other, with all else held equal. The same is true with /ɛɪ/ versus /ɛ/.
