I can't say much about Carthage or Athens, but for Rome I know the fall of King Tarquin to be more of a legend than recorded fact.

There are no records for the traditional "Seven Kings of Rome" due to a Gallic sack of Rome in 390 BC. There could have been exactly 7 kings during the lifetime of the Roman monarchy, but the likelihood of seven consecutive kings presiding over long and eventful rules in the ancient world seems unlikely. The Seven Kings story seems to be a continuation of the Romulus/Remus founding myth of Rome, acting as a way to explain each founding principle of Roman civilization.

For example, Romulus created Rome as a militaristic society, but his successor Numa Pompilius tempered that militarism with religiosity and spiritualism. This explained Rome's emphasis on organized warfare and simultaneous devotion to spiritual contemplation. Noticeably, warfare still came first. This neatly explained the origin of the Romans as a militaristic people tempered by religion.

Similarly, the final king of Rome, Lucius Tarquinius Superbus, was meant to explain the Roman distaste for absolute power and his tyranny served to remind the Romans why they created the Senate in the first place. This reminder worked especially well when the slum-dwelling lower classes heard of the fat and greedy Senators lounging about in their country villas.

This was a long-winded way of me saying that the "exact date" of 510 is not the exact date of the founding of the republic, being more of a ballpark range. Recorded Roman history is far more accurate and well documented after 390 BC. While the connection between Carthaginian and Athenian transitions to republics/democracies might be a bit more than coincidental, the exact date of the founding of the Roman republic is unknown, and Rome's transition to a republic, and any relation that had to other mediterranean societies, is unknown.

I think the premise here is pretty flawed for a number of reasons, many of which have to do with our patchy knowledge of Ancient history.

First, Antiquity is a long period. It's longer than any other conventional period of history. It's mostly defined as the 3500 years from the invention of writing to the fall of the Western Roman Empire. If, within this vast stretch of time, certain things seem to happen very quickly, we always have to ask if we'd have said the same if those events took place in another era. The constitutional changes you're talking about supposedly happened between 594 BC (the reforms of Solon) and 480 BC (the Carthaginian defeat at Himera). That's 114 years. That's the time between Waterloo and the Great Depression. Does it really make sense to consider this some sort of 'Mediterranean revolution'?

Second, even within this actually vast period, you're already massaging the examples to make them fit the same mould. Athens did not transition from monarchy. The kings of Athens are mythical; there is no historical period in which Athens had kings. As far as we can tell, Athens at the time of Solon was ruled by a closed aristocracy of wealthy families known as the eupatridai ("those of good fathers"). The sum effect of his reforms was to open up this aristocracy somewhat and to improve the plight of the poorer classes that suffered under it. But soon there was a backlash, and Athens fell into tyranny, and it's only after that (with the reforms of Kleisthenes in 508/7 BC) that we start the gradual change to democracy. But most historians argue that it took until about 460 BC before this process was complete. The pseudo-Aristotelian Constitution of Athens lays out for us how twisty the road to democracy was at Athens, and how the version we know best really belongs to the period of the restored democracy after 403 BC. (As /u/A_Village_Idiot_ points out in this thread, the situation in Rome is also a lot more complex than simply "they overthrew the kings and became a republic at this precise date".)

Third, there's no reason to believe that each of these movements to create governments on a wider franchise go back to a common Greek example. The forms of government that emerged in Greece, Rome and Carthage were totally different; the fact that they weren't monarchies is pretty much the only thing they had in common. If we are to suppose that the Roman republic was inspired by Athenian democracy, we'd have to explain why these systems have practically no similarities in terms of their institutions, their political classes, their daily operation, or even their definition of citizenship. Recent scholarship mostly assumes that the Roman political system is an indigenous development, at best informed by contacts with the local Greek communities (none of which were founded by Athenians until the mid-5th century BC), but not based on anything Greek. This is laid out in detail by Michael Fronda in Dean Hammer's Blackwell Companion to Ancient Democracies and Republics (2014).

Fourth, it doesn't explain much to try and reduce all these developments to inspiration drawn from a Greek idea, since we don't really know where the Greek idea of democracy comes from. Many books have been written on just that question. We can find traces of democratic ideas in Greek writings as early as Homer, and several states seem to have had conciliatory political institutions long before the foundation of Athenian democracy (in Democracy: A Life (2016), Paul Cartledge lists Megara, Mytilene, Cyrene, Herakleia Pontos, Naxos, Eretria, Chalkis, Lipara and Ambrakia). But it's nearly impossible to trace the development of an idea of political equality, or to figure out where such an idea might have come from. What we can say is that the idea of non-monarchic rule was not exclusively Greek; the Phoenician cities also had ruling councils, which is the much more likely origin of the Carthaginian system. Nor was it universal in the Greek world, since many Greek states were run by tyrants or royal dynasties, and there is no clear trend for their number to recede over time, as shown by the evidence collected in David Teegarden's Death to Tyrants! Ancient Greek Democracy and the Struggle Against Tyranny (2013).

In short, I don't think this question really works. The period between these events is actually pretty long; the events themselves are not the same; the resulting government systems aren't comparable; there's no reason to assume they share an origin; the origin is in each case actually pretty obscure. When we look at the history of each state in detail, we can see that these moments on their timeline aren't actually that easily seen as a Mediterranean trend.

Adding on to what others have already written, I can comment a bit more specifically on the Carthaginian side of things.

In essence, Carthage never seems to have had its own monarchy. We have no references to kings in any surviving Punic inscription, and while Greek and Roman authors do mention Carthaginian "kings," they also generally include some sort of qualification, noting for example that the "kings" ruled in pairs and that they owed their political power to constitutional or electoral processes. The use of "kings" here thus seems to approximate the Phoenician-Punic title špṭm, which literally means "judges" and which designated a community's chief civil magistrates.

Prior to its emergence as an independent state (perhaps in the late-sixth or early-fifth century B.C.E.), Carthage presumably came under the authority of the King of Tyre, a royal governor, and the local council/senate, as suggested by the close ties between Carthage and her mother city as well as situations documented in other Phoenician colonies. The council/senate in particular would have provided a pre-existing foundation for the republic. Unfortunately, we know too little about early Carthaginian history to speculate much further. :(

Sources and further reading:

Ahmed Ferjaoui, Recherches sur les relations entre l'Orient phénicien et Carthage, Orbis Biblicus et Orientalis 124 (Fribourg: Editions universitaires Fribourg Suisse; Göttingen: Vandenhoeck & Ruprecht, 1993), 40f., 172-175.

Maurice Sznycer, "Le problème de la royauté dans le monde punique," Bulletin Archéologique du Comité des Travaux Historiques et Scientifiques 17, fasc. B (1981): 291-301.