In the world of pets both tattoos and chips are already widely used as a means of marking animals so that if it is somehow lost to the owner it can be recognised and brought back.
Just to make it very clear - whatever method is used it contains merely an identifier that is linked to a database entries. Any solution where the agent (tattoo or chip) should contain also the data requiring updates is prone to too many errors and problems.
As far as I can see the general direction is to use chips rather than tattoos. And there are few good reasons for that:
1. Implanting a chip is far easier, less invasive and time consuming than putting a tattoo. I guess everyone knows these days how much trouble it is to make even a simple tattoo. Implanting a chip is a single simple injection with a chip itself being a bit larger than a single rice grain. Also making a tattoo is rather painful. Again, implantation is simply an injection.
2. The chip lasts longer. Since it is using an RFID technology it does not require any power (the power is induced from a scanner to send back a signal). There is hardly anything that can break or fail. If it works it'll work for years and years. And it will always produce the same return signal. 75
The tattoo is exposed to all the risk the skin is. It fades over time, gets blurry as the ink dissolves to some level, might become unreadable due to scars, burns etc. As a result it might even produce a different value than it had initially. Of course you might do a correction to a tattoo or even make a new one if the old is entirely unreadable but that brings us back to point one ;-)
3. Skin imperfections might make it difficult to effectively place a tattoo in a specific skin location. If you have a scar you can cover it with a tattoo of course but it won't be a precise one with sharp lines (as needed for bar codes). Similarly mole might cause it impossible to blacken a specific spot. I believe you should try using the same place for the identification tattoos to avoid multiple identities of a single person. A chip might be prone to similar problems (sometimes - but rarely it can actually even shift its position) but it's far easier to scan entire body with a proper scanner than to scan entire skin.
4. Chip resides under a skin, thus is invisible so it's less obvious. Of course it can work both as a benefit and a drawback depending on the goal to be achieved. But still if we're considering more humane option, I would consider something invisible better here.
Note, I'm not referring to artificial identification versus biometrics here as that is not what the OP has asked for. I'm also referring to the currently existing technology.
And guess what, not only the technology is in place, it has also already been enabled to people who wants to use it and are lucky enough (?) to have it within their range. Some reading to find out more.
BTW, all my pets have chips implanted ;-) 225