Many aspects come down to an "it depends". For instance:
What are the circumstances of their arrival?
Did they arrive on Earth by way of a freak accident? Or were they deliberately sent? For instance, as an ambassador to our reality, or maybe as an 'astronaut' who's testing out a new technology? Or is hopping into an alternative reality just how their people take vacations?
The more deliberate their arrival, the more opportunity there was for forethought, and the more likely it will be that they'll have something in their pocket that's tailor-made to prove their identity (or at least, their alienness). For instance, an astronaut/experimental scientist is likely to have a recording that documents their entire journey. Someone being intentionally sent as an ambassador will have whatever their best scientists (and/or politicians) think will be most convincing (and if they've done this sort of thing before, they probably have a good idea of what's most convincing).
Someone transported by accident will have the hardest time, as they'll only have whatever items were on them at the time of the accident. So essentially, whatever a person might typically have in their pockets. Maybe the alien version of a smartphone is unquestionably more advanced than what people carry around on Earth (and hopefully never requires a recharge)?
Are we talking alternate-reality humans, or aliens?
An alien life-form has an advantage in that it can easily prove that it's not just another undocumented/fraudulently documented human being, by virtue of its alien physiology. It can reveal this at the appropriate time/to the appropriate people to prove its identity as a bonafide alien visitor.
A human from an alternate reality would have a more difficult time, as our border officials already have to deal with unknown humans possessing no or strange documentation. There are procedures in place for that already, and if your traveler can't produce compelling evidence of their alien status then those procedures will likely be followed. They'll be detained, interrogated, and ultimately deported to wherever the border officials think they came from (or else held indefinitely/possibly shipped off to an insane asylum if instead of saying where they came from they keep insisting that they're an alien visitor). Their human biology works against them.
Where are they arriving?
Does your traveler materialize in the middle of the customs line at LAX, where people are standing by and prepared to deal with (and question/scrutinize) new arrivals from foreign lands? Or do they arrive in the middle of a field somewhere, where nobody is going to see and nobody is going to go asking after their papers until they try to do something that would require them? Is their arrival caught on film, providing evidence that they basically appeared out of thin air?
Similarly, it can also make a difference if they arrive in a nation with strict immigration/border protection policies such as the U.S., as opposed to a more relaxed nation like Fiji or Samoa or similar small island nation with a cultural history of inclusiveness. The former is likely to present them with more hurdles than the latter.
Anyways:
How would they use their native documentation to obtain a legal set of identity and travel documents for our reality?
I don't think they do, unless the native documents have some intrinsic value in terms of proving their status as an alien traveler. As in, they don't just say "John is a citizen of Reality 123Q", but they say it using a method or technology that's so outlandish or advanced from what's being used anywhere on Earth that it lends credence to their story about being from Reality 123Q.
Apart from that one use (and perhaps sentimental value), their native documents are worthless. They can't use them to go back home. And nobody on Earth can authenticate them, because nobody on Earth can get in contact with the issuer to determine what the authentication procedure actually is. It doesn't matter if they're genuine or not, because nobody can tell the difference anyways. The documents are only useful to the extent that they're alien enough to lend prima facie support to the traveler's claims of being from an alternate reality.
Essentially they're down to this:
They can reveal that they are from an alternate reality/alien
planet/... if needed
They'll need to make that revelation, at least to one person with enough power or connections within the relevant government bureaucracy to get some local documents for them. If they're unable to do so convincingly, then at best they'll be dismissed/ignored and at worst they'll be subjected to all sorts of scrutiny (and probably also involuntary detention) that they do not want. It pays to befriend this person first to help mitigate the possibility of those negative outcomes.
An alternative would be to seek out forged local documents on the local black market. I tend to think that this is what the savvy cross-dimensional traveler might do, especially if they want to keep a low profile (which they might unless they've been sent as an ambassador). It's much lower risk, especially if they don't have a clear way of proving their alien status (such as an alien physiology). And then it's just a question of how good their forged documents are, and whether or not it's possible for illegal aliens in their locale to gain amnesty with a path to legitimate citizenship. If that path is available, it's probably the safest one to walk.
