Actually I would remove it. Any sitting water is going to get thru the stucco and rot the framing...sorry.
I've had to re-stucco any area cuz of former owner doing something like that but there was no rot.

Whatever you put in will have to be small and will need water due to lack of depth for roots. I don't think 6 inches deep is enough to hold any water. A tray might help and that would partially eliminate the rotting issue.
Ivy geraniums would look nice but 6 inches isn't enough.

Actually, if that was me, I would get some drought tolerant plants, maybe the ivy geranium, grasses, succulents for draping over. Do not plant them in the container, just leave them in pots. Then you can change them out when they don't look so good, or for example, with the holiday season here, swap your plants out for poinsettias in cans, and when those are done remove them and put in spring bulbs in cans, and on and on.

No issues with rotting out the stucco with this metal planter box that is sitting on top of the ledge, as long as you provide an air gap between the bottom of planter and stucco cap. I would suggest sticking with lower water using succulents for this area, and things like Graptoveria 'Fred Ives' Graptopetalum paraguayense, Kalanchoe blossfeldiana, Aeonium 'Swartzkop', Crassula erosula 'Campfire' and Cotyledon orbiculata and possibly some of the trailing Iceplants such as Lampranthus spectabilis or Oscularia deltoides, as these would all thrive in this planter with a drip irrigation system hook-up timed for just a minute or two as needed. These plants would combine to grow up and also drape over, and would give some color/texture year round without needing to be changed out. I'd suggest that you add thin bricks as a spacer below the planter bottom, and add time release fertilizer into the planting mix before you plant, and top off every 3 to 4 months with additional time release fertilizer.

Yes, succulents. Very little care required, and if you forget to water them they won't mind. Echeveria 'After Glow', Calandrinia grandiflora, Echeveria imbricata, Kalenchoe luciae, Sedums, and so on. You get amazing foliage color all the time. If you switch out the potting soil every year or two, fertilizer is not necessary. Give them a spray of rubbing alcohol or soapy water if you get mealie bugs.

The thing that concerns me is the mid-summer sun. Echeverias would look lovely, but they might burn up in the summer, depending on how hot it is there. Mine get cooked in the summer, so I had to move them all to the shade. Aeoniums get huge, and they can burn as well.

I would buy needle- leaf or variegated ivies and leave them in their pots (6") and drop them into the planter. In the winter, you can drop red cyclamens in the four inch pots in between the ivies, and when they go dormant in the spring, you can substitute petunias or another colorful bedding annual like portulaca.

Another option is Sprenger's asparagus ferns- but beware. The roots expand and you can't get the dang things out without breaking your pot. They drop berries which turn into plants within nanoseconds. I keep mine in plastic pots for this reason. These tough plants will take either full shade or blazing sun.

It's just tough to find anything that will thrive in full shade for half the year and full sun for the hot summer months, especially if its roots are being cooked in a black planter. My dad had this problem on his patio- he bought gorgeous (expensive) hand-forged rusted-metal hanging pots, and that summer every plant was cooked in its own little cauldron. Boo.

Renee