Can I give territory to an ally?

Giving territory to an ally would be especially useful in the level Break the Bank: the economics in that level make it hugely beneficial for your ally to be as strong as you.
Additionally, the ally could get a three armies income boost by owning a province where the player starts. Gifting this province to the ally would bring it up to par with the player, and boost total income by 3.

Yes, with the Gift card

The gift card lets you do exactly that. If the card is not enabled, for example like in the scenario you linked, you can instead let your ally take the territory from adjacent spaces by having them ensure the relevant attack/transfer is set to 'treat teammates as enemies' instead of normal. The AI notably is terrible at this and will never attack teammates, even to take a bonus. You, however, certainly can take some of its spaces for your own, which may be useful even if it's not entirely ideal since you can then funnel troops to the AI by transferring them through that territory. The AI will move any troops you give it that way up to its front in a pretty reasonable manner.