Why is Severus Snape's Patronus the same as Lily Potter's? Are Snape and Lily related somehow? Are Snape and Harry related?
They are not related, it is Snape's profound love for Lily that causes his Patronus to be a 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.
I don't think it is official known if Snape's Patronus changed at any point, however his happiest memories even from childhood were that of Lily Evans. Since the Patronus requires happy thoughts, it is only natural for Severus to draw on these.
"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
Snape’s matches Lily’s as he loved her.
The reason that Snape’s Patronus matches Lily’s is because he was in love with her, as Harry tells the Dark Lord.
“You never saw Snape cast a Patronus, did you, Riddle?’
Voldemort did not answer. They continued to circle each other like wolves about to tear each other apart.
‘Snape’s Patronus was a doe,’ said Harry, ‘the same as my mother’s, because he loved her for nearly all of his life, from the time when they were children.”
- Harry Potter and the Deathly Hallows, Chapter 36 (The Flaw in the Plan)
His Patronus is the same as hers because of his feelings for her - it doesn’t mean they’re related in any way.