One verb, two objects
The verse extends a single act of knowing — "that they should know you" — to cover both the Father and the Son with one conjunction. It does not say 'and also to know him.' The same verb simply stretches to hold both.
Grammar can be as precise as a legal document or as open as a handed key that fits more than one lock.
Here the structure is: know [you] and [him whom you sent]. One verb, two objects joined by 'and.' In ordinary speech this usually means the same act extends to both: 'I know the owner and the manager' does not normally imply two different kinds of knowing. The knowing simply reaches further.
But the two objects here are not equivalent in the way an owner and a manager are. One is addressed directly as 'the only true God.' The other is introduced as the one 'sent' — a different posture entirely. The verse gives both to the same verb without pausing to say whether the knowing works identically in both directions, or whether the same word covers two different relationships the way a coat can cover two different shoulders.
The grammar neither collapses the distinction nor marks it. It simply places them together and moves on.