Disclaimer: I'm not a biologist.
Think of DNA as source code for programming. It is read out and executed, thus very important. So if you just add some ID on it, if in your story DNA is not completely understand, it could be very easy to destroy something, just like when you randomly insert a line of code in a software project with million lines of code. Nothing could be changed, or everything could go to hell.
(EDIT: Junk DNA was mentioned in the comments. It would be a reasonable place for IDs, given you can place it there. As I understand from the Wikipedia article, biologists are still guessing why it exists in the first place, it may as well be important for growth before birth. Money quote: "Several lines of evidence indicate that some "junk DNA" sequences are likely to have unidentified functional activity")
Also DNA changes over time. Due to different factors like radiation, it is altered and doesn't reproduce like earlier, but with errors. Most obvious example of this is aging, but tumors are also one. So it could as well be that by "reading out" the ID, a wrong ID is read out cause it was altered.
Here is an idea how you could do it instead: Write the ID inside their skulls. Take out a part of the skull bone, a random unique part, carve the number on the inside in, and sew everything together again.
- Clearly permanent.
- Difficult to temper with: A single bone can normally replaced by a similar one without the body feeling noticing. But a brain operation is delicate. And since the part of the skull bone is selected random, to temper with it you would have to replicate that exact part (or people could see there are several traces on the skull bone, for which they would only have to open the head, but not the skull). For that, you first need to take it out of the skull to take measurements. For the time being the clone would lie around with exposed brain. A lot of difficulties and problems. The drawback is that it is difficult to carve the number in the first place, but still a lot easier than changing it.
- As long as the operation is done by expert in a sterile environment, there would be no consequences but a scar on the head, usually covered by hair.
Ok, right before posting I notice it would also be very difficult to check the ID (as long as the clone needs to stay alive), but you can circumvent this by encarving the ID on both sides of the skull bone part. Easy check and hard check possible. Another variant would be to only write it on the outside, much less dangerous because no brain exposed (because operations always have a risk), but then again easier to temper with, the random unique traces of the removed skull bone part would be missing.
EDIT: Another way of checking the encarvings would be a MRT. Giving technology for clones is available, it is reasonable that scanning the skull bone is mobile possible too. Today MRTs are done in clinics and can take e.g. half an hour, even that would be reasonable. And if the encarving is filled with some metal e.g. iron and a protective silicon layer so the metal doesn't interact with the rest of the head, this would make checking even easier.