I have a open loft apartment. The ceilings are 25 feet high. I am
trying to divide space for my five year old daughter so that it gives
her some privacy.
One suggestion would be to use bookshelves. But because they would be
at right angles to the walls, I don't see how I could anchor them.
If they're at right angles to the walls, you securely attach the end to the wall, top and bottom, choose an adequately rigid design, and bolt them together top and bottom. I'm guessing you're trying to avoid screwing to the floor and you're OK with semipermanent, you can augment with double sided tape or adhesive to the floor, which if done right will be an absolute PITA to remove, but will still be removable without damage. I wouldn't count on double sided tape as a sole solution but if you're OK with permanent, there are adhesives that would do the job. If they need to be freestanding, bolting multiple shelves together in an L or T shape can be adequately safe provided they are weighted sensibly.
I'd prefer not to use panel room dividers, but I am not sure what is
another option that is safe while also aesthetically pleasing.
Actual walls come to mind. If you build a wall tall enough to build a door into it, you can use the L shape principle mentioned above and secure both ends of the door to walls, producing safety and strength for the price of stud and drywall, which is probably much cheaper than bookshelves. It would also give you something to screw your bookshelves to. Again, if built correctly, they don't necessarily need to be screwed into or adhered to the floor.
So my questions are:
Is there a safe way to place bookshelves in an open space so that they do not tip over?
Ooopsie doodle I kind of answered before the question, but aside from the options mentioned above, use a bookshelf design with a wide enough base and low enough center of mass that it can't plausibly be tipped (Like back to back 18" deep 6 foot high solid wood shelves with textbooks on the bottom two shelves, or a single shelf with base extensions.
Do you have a suggestion for dividing a room other than panel room dividers?
You seem to care about finish, and some of the panel room dividers you're asking about can be quite fancy, soooooo... Maybe it would be better to say what you do want it to look like. I recommend actual walls because cost is low and finish is arbitrarily good, but if you can't get a building permit and you're not allowed to build legal "temporary" walls, the extra cost of bookshelf walls may be worthwhile.
NPGB but no need to add this in your question.