Are these actually two different dolls?

No. Both are the Barbie Signature Holiday Sweater doll, product number JML11, in the blue Hanukkah outfit. The listings describe the same doll with different wording.

Which is better for a young child?

They are the same doll, so either works. The Mattel listing states a 6Y+ age rating, which helps if you are buying for a child around that age.

Does the light-up feature work the same on both?

Yes. The sweater lights up with a button press on both, and the menorah branches are lit. Owners say the feature works as expected.

Which one is more likely to arrive undamaged?

Neither listing guarantees it. Owners of both report crushed boxes from bag-only shipping, so confirm the seller ships in a box with padding before buying.

Is the box worth keeping for a collector?

If you collect in-box, box condition matters, and that is the main reported flaw. Ask the seller for photos of the actual box before you commit.

Barbie Hanukkah Doll vs Mattel Hanukkah Barbie

These are the same doll: both carry product number JML11, the 11-inch Barbie Signature Holiday Sweater doll in the cobalt blue Hanukkah outfit with the light-up menorah sweater. The listings differ in how they describe it, not in what you get. Buy whichever listing has the doll in hand and intact packaging, since crushed boxes are the one complaint owners raise. For a gift, the box condition matters more than the listing.

Choose the Barbie Hanukkah Doll if that listing confirms the doll is new in an unopened box and the seller ships it boxed with padding.
Choose the Mattel Hanukkah Barbie if that listing shows the doll and box clearly and the seller packs it in a box rather than a bag.

Prices

Barbie Hanukkah Doll: $50.00 at Mercari (checked today)
Mattel Hanukkah Barbie: $50.00 at Mercari (last seen Nov 9)
Difference: both cost the same at the latest prices (Nov 9 and Nov 11)