I've seen this before. I can't tell if this is limestone from this picture but it most likley is. We have a demolition service and thats where I encountered this problem before. In our case, and it looks to me like this is the same situation, the installer put 9 dots of thinset on the back of each tile instead of back buttering the entire thing. Also in our case, and most likely in this case as well, he (assuming gender...not politically correct...sorry...not really) didn't even bother to put thinset on the concrete (assuming it is a concrete slab). If he had there would have been a more uniform staining and maybe nobody would have noticed.
It's likely poorly bonded to the concrete. I'd rip it out before I tried to fix it but then again, not everybody has a demo service. Having said that, I agree with Baird. I would probably try a good enhancer but managing expectations would be my priority.
Jeff Hammons
Managing Partner
Cutting Edge Flooring Services LLC
dba Cutting Edge Surface Pros
dba Cutting Edge Concrete
dba Cutting Edge Tile Removal - Certified DustRam Contractor
4125 Hollister Suite P
Houston, TX 77080
Phone: 281-704-2819