Is there some kind of share spell option for the paladin mount?

[Myth27]

What the title says

[Skelechicken]

Kind of. Using find steed means as long as the Paladin is actively on the mount, only self-targeted (only) spell applies to the mount as well. This is pretty limited in scope, but as far as I am aware it satisfies the definition of "share spell option."

[Malifice]

What the title says

Yes. It's literally mentioned in the spell description.

Read the PHB.

[Myth27]

Ok I missed that

“While mounted on your steed, you can make any spell you cast that Targets only you also target your steed.“

But I still don’t understand if it means The paladin can cast “self spells” on the horse or if it means a single spell can affect both the paladin and the horse

[Nettlekid]

Ok I missed that

“While mounted on your steed, you can make any spell you cast that Targets only you also target your steed.“

But I still don’t understand if it means The paladin can cast “self spells” on the horse or if it means a single spell can affect both the paladin and the horse

Everything you need is in what you just quoted. If you cast a spell that targets you and only you, you can make it target your steed as well. Nothing else, which means it doesn't give you any ability to cast a spell that would normally affect you and instead have it affect your steed and not you.