My Shimano Ultegra ST-6510 shifter name plate broke and then fell off.
The other one was cracked as well but didn't fall all the way off. The rest of the left STI shifter was fine - I don't use the front derailleur too much. The right one had some funny issues where you have to hold one of the levers as you shift, but I made it work.
Anyway, regarding that name plate: I couldn't easily find a replacement online or in local shops because Shimano doesn't make them anymore. So I decided to 3D print a custom name plate. I wanted one for cosmetic reasons and to prevent dirt and water and so on from getting in there as I was already having some issues with shifting. Also, an Etsy customer was having a similar issue and helpfully nudged me to get this done.
These should also fit the 5510 and R600 style levers also, and obviously are not an official Shimano product. These ST-6510/5510/R600 name plates are now for sale at a modest price in my web store.
I did a bunch of solid modeling by trial and error. It took about a dozen tries to get things to look and fit right, but I'm happy with the results. I don't yet have replacement screws available but usually these are still attached - it's just the name plate that cracks and then eventually falls off as the plastic cracks even more and the screw can no longer hold the plate.