"My Mother Told Me" is a famous ancient Norse Skaldic song composed by Egill Skallagrímsson.
Egil wrote skaldic poetry, and that is why in the Old Icelandic manuscripts containing a version of "Egil's Saga", the skaldic verses are quite distinct in style from the prose.
Some researchers consider that Snorri has written the Egil Saga entirely, while others do not, although it is certainly possible.
All of which is to say that the ages depicted in the series are mostly the 9th, Egil's Saga is from the 13th, although it could of course have genuinely early oral content.
The skaldic poetry that it contains could be truly ancient (around the 10th century), as well as Ragnarsdrapa, which speaks, of course, of Ragnar Lothbrok, and other poems.
Skaldic poetry has not disappeared in the Viking era, Snorri has made sure of that.