The “snake”, as you put it, is Anantashesha - the Oldest Son of Kadru and the Oldest non-humanoid demigod to be born… and the Patron God of all Dragons.
Through extreme meditation and soul-reflection, Anantashesha achieved Parama-Moksha (Great Liberation), and reached Vaikuntha - the highest heaven. Here, he proclaimed his devotion to Sri Vishnu, who graciously took him as his Regent of Balance.
Anantashesha name literally means “The Unending One” - as in when the Brahma-Pralaya (Omniversal Death) occurs and the Mahadeva ends the Omniverse by lighting it awash with the Ultimate Reality, Anantashesha will the only one who’ll survive the End of the Omniverse.
Anantashesha is an Elder God (the second highest tier of Godhood, after Supreme Gods). He’s among Lord Vishnu’s loyalest retainers and Regent (Deputy) to the Mantle of Omniversal Balance.
Lord Vishnu reclining or sitting atop his coils symbolize both how Lord Vishnu reigns greater than Eternity itself, and how much Anantashesha considers himself honored by being the vehicle and seat for Lord Vishnu to recline as he constantly meditates the Omniverse into a state of Balance.
Anantashesha’s also has infinite number of heads - each head symbolizes a different parallel universe. So Lord Vishnu reclining atop the Anantashesha also symbolizes his mastery over Infinity.