[RPG] If a spellcaster forgets Find Familiar, can they keep the familiar

Straight from the DMG Errata:

Attunement (p. 136)

The first paragraph ends with a new sentence: “If the prerequisite is to be a spellcaster, a creature qualifies if it can cast at least one spell using its traits or features, not using a magic item or the like.”

emphasis added

Therefore, for the purpose of attunement:

If a creature is able to cast a spell by virtue of their own ability (not from an item), whether that be race, class, or feature, you are a spellcaster.

To start with, your familiar can't use Shocking Grasp. Find Familiar says:

Finally, when you cast a spell with a range of touch, your familiar can deliver the spell as if it had cast the spell. Your familiar must be within 100 feet of you, and it must use its reaction to deliver the spell when you cast it.

So you use Shocking Grasp, and your familiar merely delivers it. This all happens on your turn, too, not your familiar's turn.

However, this is a good thing for you, because Thunderbolt Strike says that:

At 6th level, when you deal lightning damage to a Large or smaller creature, you can also push it up to 10 feet away from you.

So the fact that you cast Shocking Grasp, and you damaged a creature, means that you can indeed use Thunderbolt Strike to push it away from you.

Note that that's away from you, not away from your familiar: Your familiar delivers the spell, but you still cast it, and you dealt the damage. If that wasn't the case, you wouldn't be able to use Thunderbolt Strike at all.