If you look at mainstream physics multiverse interpretations such as the Many-worlds interpretation (MWI), you can get a lot of ideas from the current scientific hypothesis.
Unfortunately there are a number of challenges your fictional universal graffiti artists (a.k.a. "taggers") will face:
Other "you" comes up with the same idea
A major problem with "tagging" your universe, however, lies in this:
Paul Steinhardt has famously argued that no experiment can rule out a theory if it provides for all possible outcomes.[5]
The MWI (as well as some other similar multiverse models, even expanded universe models (c.f. Hubble volumes)) implies infinite, or at the very least, uncountable universes. What happens when a possibly infinite group of other "you"s come up with the exact same idea, and even the exact same identifier for their universe?
My suggestion for your Homing Device
As others have already suggested, but I hope I'll be able to provide much more detail about, is quantum entangled objects. This is a complex and subtle topic, so it bears some explanation.
It's possible to use some sort of quantum entanglement between a device you carry with you, and a device you leave at home. The key here is that these devices must be created ahead of time—if your device is lost or stolen, you will be completely, possibly permanently cut off from your universe!, which I expect would bring up some interesting potential sub-plots.
These objects also have a lifetime and are very sensitive to interference. So far, we've only been able to keep quantum entangled experiments running for a few minutes to a few hours in very carefully controlled tests. Even accounting for significant advances in science, this device—perhaps second only to your ship's life support system in terms of importance—would likely be rather delicate and sensitive.
Quantum entanglement (QE) and superpositions can't be used directly for your homing device, since quantum entanglement cannot actually propagate information over distances! Read that again. That would violate causality in a pretty big way. You cannot use this device for communication.
So what's it good for then? QE essentially guarantees you and your favorite Earth will receive the same stream of random data. Happily, it is resistant to eavesdropping—as a consequence of physics, no one else will be able to intercept that random data. To be useful, combine the QE with quantum key distribution.
You and Earth then take that random data, converted to ordinary binary bits, and obtain a unique shared secret (like a password) that both of you know. At this point, you must then use some ordinary (non-quantum) inter-dimensional communication channels (your creative skills will be required there).
Your shared secret can be used with the one-time pad, (OTP) which will effectively guarantee you're talking to the desired Earth. In the real world, OTP has been rigorously proven unbreakable when used correctly, and it is absolutely the right tool for the job in this situation.
"Bonus" question: tagging more than one universe
Absolutely. You can simply extend my answer to include multiple pairs of entangled homing devices. One end stays on your Earth (or maybe your spaceship, if that's your base of exploration?) and the other end is planted in the desired universe.
If you tag many universes, storage may become an issue, since a pair of physical devices is required for each universe. In that case, you could instead create a network of paired devices, similar to peer-to-peer computer networks. In general, graph theory will give you the building blocks to come up with a decent fictional entanglement network.
How big are these things?
I don't have a good answer for that, unfortunately. Our experiments to date have not really been concerned with size (one of the successful experiments required a particle accelerator!) The actual entangled matter would be just a small group of subatomic particles, but even with a futuristic advanced fictional version, I'd expect it would need some kind of powered containment vessel, plenty of shielding, and plenty of redundancy.
Personally, I wouldn't go any smaller than a coffee mug, but if you want to make it central to the story, you have a lot of freedom on this point. Make it small enough to steal? Big enough to require a lot of power? Smaller and more vulnerable? Bigger, but less room for more of them?
What have we learned?
- Quantum entanglement must be done before you leave home, and the devices must be kept safe, or communication will be impossible, given infinite (or just plain "many") worlds
- Quantum key distribution basically gives you a long "password" to prove it's you
- One-time pad encryption/decryption is what you would use to actually send/receive a message, but you must use more conventional (non-quantum) communication methods
Why is this all necessary?
(Further discussion, since MWI/multiverse presents a big problem where identification is concerned.)
While you may have nefarious villains (or desperate good guys) who would try to trick you, get you to fly your valuable ship into their universe, etc., the real problem is just that there are so many universes. There might be an infinite number of universes where you cut yourself shaving that morning. And an infinite number where you didn't.
Even if your number of "reachable" universes only lies in the dozens that are similar enough to warrant this question you asked, that's still enough to preclude any sort of human-knowledge-based tagging system.
Say you generate a random tag, "puppies56831X", and call it a day. Some alternate you (even infinite alternate yous), might be similar enough to generate the same random tag and otherwise be indistinguishable under close scrutiny.
Quantum superpositions are the only potentially valid answer I know of, which is why I have focused my answer on them even though there is at least one other answer that mentions them. I did not get the idea from that answer, nor did I borrow any of the specifics—I barely skimmed it. I just hope my treatment of the topic, as it relates to your question, is helpful to you.
