Yes I second the Command Hooks. I think you have to let them sit for 24 hours before you use them to really make the glue stuff stick. My daughter used the heavier duty one to hang her backpack all through college, so I'm sure they're strong enough for a robe.

3M hooks depend on the finish of the surface, too, though. If the doors are gloss or semi-gloss they may not be sticking as well. I always clean the surface I want to stick the hooks to very well with some rubbing alcohol, let that dry, then stick and leave it a good 24 hours before using. If the surface is glossy at all I would use some sandpaper to LIGHTLY rough up the surface before cleaning. (Just enough to give the adhesive a better surface to stick to.)

If you don't want to do that, I think the nicest look would be to get someone to make you a wooden plaque of the appropriate length to bridge from the sections thick enough to screw into, with the edges finished in a way complimentary to the trim style on the doors themselves. (I.e. The edges of the plaque routed to match the profile of the door trim or similar.) Then paint the wood the same color as the doors and fix hooks to it - for this look I'd do maybe 3 hooks, spaced nicely apart, not just one lone hook in the middle - then fasten the plaque-with-hooks to the door and touch up as needed to conceal the fastening screws.

If you get a talented trim carpenter to do it, they could hopefully add the plaque for hooks in such a way that it looked an intentional part of the door design, rather than a bolted-on after thought. But 3M hooks would definitely be cheaper. :)

Is your door thick enough at the top to do an over-the-door style hook, such as the one pictured attached? There are many available on Amazon, and elsewhere.

I like the over the door type robe hooks they don’t damage the door and hold whatever you want

the over the door ones show from the other side, though. which bugs me unreasonable amounts. :)

We have the same style of doors - we love them! We love them so much we bit the bullet and had replicas made for some mismatched doors that weren't original to the house.

To your question, though: In the original bath a previous owner mounted two hooks - one on either side in the 2nd thickest part of the door. We added a double robe hook to the back of the guest room door but we mounted it up on the thickest part of the door (centred).