De är inte relaterade, det är Snapes djupa kärlek till Lily som gör att hans patron blir en doe.

The Patronus, asserted Spangle, represents that which is hidden, unknown but necessary within the personality.

‘For it is evident,’ he writes, in his masterwork ‘Charms of Defence and Deterrence‘… that a human confronted with inhuman evil, such as the Dementor, must draw upon resources he or she may never have needed, and the Patronus is the awakened secret self that lies dormant until needed, but which must now be brought to light...’

...

The form of a Patronus may change during the course of a witch or wizard’s life. Instances have been known of the form of the Patronus transforming due to bereavement,falling in love or profound shifts in a person’s character. Thus Nymphadora Tonks’s Patronus changes from a jack rabbit to a wolf (not a werewolf) when she falls in love with Remus Lupin. Some witches and wizards may be unable to produce a Patronus at all until they have undergone some kind of psychic shock.

Patronus Charm by J.K. Rowling

Jag tror inte att det är officiellt känt om Snape's Patronus ändras när som helst, men hans lyckligaste minnen, även från barndomen, var Lily Evans. Eftersom patronen kräver glada tankar är det bara naturligt för Severus att dra på dessa.

"But this is touching, Severus" said Dumbledore seriously. "Have you grown to care for the boy, after all?"

"For him?" shouted Snape. "Expecto Patronum!"

From the tip of his wand burst the silver doe: She landed on the office floor, bounded across the office, and soared out of the window. Dumbledore watched her fly away, and as he silvery glow faded he turned back to Snape, and his eyes were full of tears.

"After all this time?"

"Always" said Snape.

Harry Potter and the Deathly Hallows