we've had a continuous cultural awareness of mammoths this entire time. even relative newcomers to siberia are aware of mammoths.
the thing is, though, that ‘modern’ (putatively post-ice age) folklore in siberian cultures tends to interpret mammoths as water spirits. because you find them (which is to say, find their remains) in wet areas like swamps.
this is not always the case. the folklore that i know best, nganasan folklore, mainly has them as something you hunt in space. you know, on the moon and such. on earth you hunt reindeer. in the afterlife you hunt lemmings. in space you hunt mammoths.
there is a lot of cultural context for this that i don't have time to get into. first of all, space is imagined as someplace you can walk or fly to, if you transform yourself into a reindeer or bird as a shaman does. sometimes it is depicted in folktales as very cold. there is a story of two shamans who competed with each other. one went to the sun to petition the sun-mother, and burned to death; the other went to the moon to petition the moon-mother, and froze to death. but it is nevertheless supposed that in ancient times, a contingent of the ancestors of the nganasans migrated to space, and live there in shining silver tents or in blinding white egg-shaped houses. it is traditionally imagined that the people who live in space are more physically powerful than those who live on earth. and that they do not speak words out loud, but use sign language or no language at all. but they nevertheless have the same lifestyle as nganasans on earth; they hunt, chasing herds. but they hunt mammoths, in addition to hunting reindeer.
they have stories of mammoth spirits, too. but i don't remember any myself. the space-dwellers are not brought up too often in folklore, nor are mammoth spirits, so overall i would rate mammoths minor characters.
i don't know what a mammoth looks like. nowadays everyone knows what a mammoth looks like. but before everyone knew, i'm not sure what they looked like. perhaps with only one eye? it is hard to say. the nenets word for 'mammoth' is related to nearby words for 'whale' and the chinese folkloric creature name qílín (whence japanese kirin). whales and qilins are both ungulates, so make of it what you will, i suppose.
you might claim this is discontinuous with impressions of mammoths as living creatures. but i disagree. the kets, of the yenisei river, have been living there since far before they were kets; they have been living in siberia since before the ice age ended, before the bering strait closed, some 12,000–14,000 years ago. they are certifiably (genetically and linguistically) related to some of the peoples that crossed over into the americas, such as the athabascans, including for example the navajos. the kets have their own folklore of mammoths; i remember their word for mammoth, tēr, but i don't know any of their folklore, so please excuse me for that.
the nganasans, who live somewhat north of the kets and whose folklore i exposited, have a somewhat more complicated history; their ethnic identity, like all samoyed identities, was created when the ancient uralic people, from the southwest, conquered and assimilated a people of considerably different ancestry (the word i know for them is the сихиртя / sikhirtya, in nenets) some 4,000 years ago. little of the folklore held by the sikhirtya survived by my reckoning, just as their language did not survive. but i believe ambient cultural impression, from the sikhirtya and the kets and so on, had an inextricable influence on folklore of mammoths nevertheless.
the ancient uralians might have had a memory of mammoths anyway; according to y-haplogroup analyses, their men originally hailed from ‘far’ eastern siberia some 8,000 years ago. but i would still consider them newcomers, as 12,000 years back, when the straight closed, they are placed in southern china. (mtdna analyses suggest a somewhat different path for the women, but it is regrettably y-haplogroups that tend to correlate with ethnic identity and language. there is no single origin the speak of on that axis.)
i am less qualified to speak on other peoples. i have heard, say, some chukchi folktales, but none about mammoths. it is unlikely they don't have them, however; i've just never looked into it.