The description of the find familiar spell states:
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. If the spell requires an attack roll, you use your attack modifier for the roll.
Does this mean that since the familiar casts the spell, it can also concentrate on it?
The initial assumption "the familiar casts the spell" is incorrect. The spell description explicitly says you cast the spell, not the familiar:
your familiar can deliver the spell
it must use its reaction to deliver the spell when you cast it
The familiar only delivers the spell, not casts it. At the moment it delivers the spell, the spell was already cast. The familiar does not concentrate on it, neither it performs any spell components. It only makes a spell melee attack, albeit using your spell attack modifier.
I think that the text in your question has the answer. I interpret the word "as if it had cast the spell" only to apply to the range of the spell.
You, as the caster, are still casting the spell, the Familiar is delivering it at touch range as if it had cast the spell. You are casting the spell, but your familiar uses its touch range. Nothing in the spell says that the familiar is the entity doing the spellcasting, you're simply delivering the spell through your familiar. Neither the spell Find Familiar nor the concentration rules (PHB pg 203) imply that your familiar can concentrate on your spell for you.
And since you're casting the spell, that means that you need to keep concentration.
