Cases like this are rare in the history of the Classic Maya, my area of expertise. I don't know much about personal unions in historic Europe, but from what I understand, they mainly happened because (1) the king of one state married the queen of another, and their child inherited both thrones, or (2) the nobility of two different states chose the same person to rule them both. Classic Maya political practices worked against both those possibilities. For one thing, queens regnant were extremely rare, and in the few cases we know of, they tended to be married to local noblemen rather than foreign kings - apparently, and not surprisingly given the small scale of Classic Maya polities, local aristocracies were strong enough relative to their monarchs that keeping them happy took precedence over any idea that royals should only marry royals.

One complicating factor is that Classic Maya political discourse doesn't really have an equivalent to the "nation" in the sense of an ethnically and linguistically homogeneous group which naturally ought to share a single government. We also don't have much evidence for the concept of a "kingdom" in the sense of a well-defined, bounded territory. (That's not to say that boundaries, and disputes about them, didn't matter!) Instead, the most important political divisions in the hieroglyphic record are among royal dynasties. Dynasties usually occupied the same capitals over many generations, but a branch of a dynasty could break off and start a new royal court somewhere else, or the principal line could pack up and move to a new capital, in a sense bringing the "kingdom" with it. Commoner farmers living in the countryside would probably have identified most strongly with their villages rather than with a larger, state-like entity, even though they might have felt that they owed fealty to a particular king.

So, on to your question! There are a few cases where kings claimed to be the kings of two distinct dynasties with different names. In one case, a king of the site of Cancuen, one Taj Chan Ahk by name, claimed to be the king of a dynasty from a second, nearby site, Machaquila. That claim probably derives from a successful war against Machaquila around A.D. 795, in which Taj Chan Ahk captured and sacrificed the Machaquila king. He might very well have taken a wife from the defeated royal family, or he might have fought the war in part because some existing family relation led him to think he had a legitimate claim to both thrones. It really isn't clear from the sources, which sadly never tell us what was on the minds of the dramatis personae. Now, both the Cancuen and the Machaquila dynasties had separate histories up to that point, and the personal union under Taj Chan Ahk did not long outlast him. By A.D. 800, his son, K'an Maax, was in power, and also claiming to be the king of both families, but in that same year, a rival king (specifically of the Machaquila dynasty) was crowned at Machaquila. It wasn't all that long before someone sacked Cancuen, slaughtered the royal family, and dumped their bodies in an ornamental pool, and, well, that was that. History doesn't relate who did it, but I see one really good candidate.

What may be an example of a much more successful personal union is attested in inscriptions from two sites, El Zotz and Yaxchilan. El Zotz is in the northern Department of Peten, Guatemala, near the much more famous site of Tikal, in the Classic Maya heartland; Yaxchilan is way over to the west, on the Usumacinta River, really near the western limits of the Classic Maya linguistic-cultural area. Rulers of El Zotz were claiming to be the "holy lords" of two different dynastic lines by the 500s A.D. Where it gets really interesting is that the kings of Yaxchilan claimed holy lordship of the same two lines. For historical reasons that would take far too much space to go into here, I'm pretty sure the El Zotz kings were using the names first, and that the Yaxchilan royal family hived off and headed west to start a new kingdom - conceivably as part of a larger plan by an alliance of kings with ties to the Central Mexican metropolis of Teotihuacan. The family names are actually better known from the Yaxchilan inscriptions, in one of which there's a "count of successors" indicating that a king of Yaxchilan was the Xth ruler in one line, but the Yth ruler in the other (I forget the actual numbers).

EDIT: Adding sources with the aim of applying for flair.

Sources:

Zender, Marc, and Joel Skidmore. 2004. "New Ballcourt Marker from Cancuen." Mesoweb. www.mesoweb.com/reports/cancuen_altar.html

I'm not sure it's exactly the same as a European-style Personal Union (mainly because I don't know much about those), but there were a few occasions where something like this happened. A famous example would be the story of the Mixtec lord/king (mixtec: iya) Nacuaa Teyusi Ñaña, usually known by the English translation of his name, Eight-Deer Jaguar Claw. Nacuaa was the first born son of the high priest of a city-state called Tilantango. It wasn't a very large center, but it was fairly regionally dominant. The priesthood in the Mixtec culture was considered nobility, but they didn't rule as kings. Instead, a lord (iya) his wife ruled the city as a pair, and the priesthood were part of the lower nobility.

Because the lord and lady ruled the city-states as a pair, this could lead to a kind of "Personal Union" situation if each of them happened to be lords of different towns. If a lady who was the heir to town A married a lord who was the heir to town B, the two towns would be politically united until the death of both the lord and the lady. You could compare that to a kind of personal union, as you could get a single person who was the ruler of two otherwise independent polities. This actually happened quite often: a successive series of marriage alliances could create very large political blocks that lasted for generations.

Nacuaa's case was somewhat different. As the son of the high priest, he was not set to inherit the throne of Tilantongo. Not content with this, Nacuaa built a reputation as a fierce war leader before setting out to found his own city-state, Tututepec. He ruled as the iya of Tututepec for a while, but then the reigning lord of his home town of Tilantongo died without any heirs. Since he had no heirs, Nacuaa, as the son of the high priest, was the next in line for succession. So Nacuaa became the ruler of both Tututepec and Tilantongo. Using this strong position, he then went out and conquered a huge chunk of the area around the Mexican state of Oaxaca. He was eventually assassinated by his nephew, and his empire fragmented after his death.

Nacuaa's story, as well as more info on the Mixtec royal dynasties, can be found in the book In the Realm of Eight Deer by Byland and Pohl.