It's close to being a one-off, and it only came about because of a unique cluster of literary precedents. The only other examples of 'historical tragedy' in 5th century Athenian drama are both by Aischylos' elder colleague Phrynichos. In the late 490s Phrynichos put on the Sack of Miletos, about the Persians destroying and completely depopulating an Ionian Greek city in 494; and the Phoenician women in 476, which like Aischylos' play was about the defeat of Xerxes' navy at Salamis. Neither play survives.
The Sack of Miletos was a disaster for Phrynichos. It was too close to home. Nearest analogy I can think of would be if some movie studio were to try and make a summer blockbuster about Pearl Harbor in 1942, instead of waiting sixty years; or if Grave of the Fireflies had come out in 1946 instead of 1988. It's possible that current events were seen as something to be dealt with in comedy, rather than tragedy, so it may also have felt like Phrynichos was making light of it. But that's only a conjecture: we know very little about comedy at that date.
And for his foray into contemporary tragedy, the Athenians voted to impose a steep fine on Phrynichos. Here's how Herodotos describes the incident (6.21):
The Athenians made clear their deep distress over the sacking of Miletos in many ways. In particular, when Phrynichos wrote his drama the Sack of Miletos and staged it, the whole theatre fell to weeping, and they fined him 1000 drachmas for reminding them of evils that were close to home. And they legislated that no one should ever stage that play again.
We can't convert ancient money into modern terms straightforwardly, but 1000 drachmas is a large amount no matter how you look at it. This must have had a huge chilling effect on the idea of 'historical tragedy'.
So Phrynichos' Phoenician women in 476 was a calculated risk. For that play, there were a couple of special circumstances. First, Themistokles himself funded the production. And this time Phrynichos switched things round: the disaster was for Xerxes, and the Athenians were the victors in the play. So he got away with it.
But it seems the 'historical tragedy' was already killed off. The Phoenician women only got written as a vanity project for Themistokles. It seems that other playwrights learned their lesson from Phrynichos' earlier bad choice with the Sack of Miletos, and they didn't try a similar thing again, except for Aischylos with the Persians.
The Persians, too, must have felt like a calculated risk. In this case Aischylos was using the same basic plot as in Phrynichos' Phoenician women, so he had that as a precedent. We can't possibly compare the two plays, since Phrynichos' plays are lost: we can't know how much Aischylos switched things up. I would presume that the second speaking role in the Persians was an innovation; if so, the dream sequence must also be an innovation; perhaps also the empathetic role given to Atossa. Also, remember tragedies came in trilogies, and the Persians was sandwiched between two mythological plays (both lost) -- Phineus, and Glaukos of Potniai. (The fourth play, the satyr play, was also mythological: a Prometheus satyr play.) We don't know what plays accompanied Phrynichos' historical tragedies.
And there's one further factor that was unique to the 470s: another kind of precedent for embedding recent events in poetic forms that were more often associated with mythology. Shortly after the battle of Plataiai in 479 BCE, the Spartans had commissioned a major ode by Simonides to commemorate it as a Spartan-led victory. Fragments of this ode were found and published in 1992. Simonides' ode is in elegiac form, with elements of epic: a formal hymnic prelude and proem; catalogue of allies; a poetic 'map' of the route taken by the allied forces. I suggest that this is likely to be a factor feeding into Themistokles' commissioning of Phrynichos' play on Salamis -- in part as an Athenian response to Simonides' ode. Even if that wasn't the direct intent, still, we're looking at a fairly unique cluster of precedents for the Persians in 472. And, it seems, later decades didn't produce any similar clusters.
Later centuries do give a wider range of themes for tragedies; one that I particularly like to bring up, because most people haven't heard of it, is a play called the Exagoge by the Jewish playwright Ezekiel (ca. 3rd-1st century BCE) on the biblical Exodus. Several large fragments survive, drawing heavily on Euripides' theatrical style; Ezekiel gives a speaking role to the Hebrew god, appearing as a burning bush. Not historical; not recent; not Athenian. But I like how it represents a diversity of literary themes in later ages.
There is indeed a significance. Very early on, there came about a sharp division between Comedy and Tragedy, with Comedy ("Old Comedy" of the Athenian 5th century) being overtly political, topical, and current in its subject, while Tragedy couched itself in myth while at the same time alluding very transparently to current events. Tragedy originally was "topical": early on, as Persians (472) shows, Tragedy depicted events of the recent past (the defeat of the Persians in 480 by the Athenian navy, e.g.). Aeschylos Persians was likely a riff on, or response to, the Phoenician Women (~476) of Phrynicus, of which we only have fragments. Both plays feature lamenting Persians, with the Phrynicus play showing the more traditional female lamenters, which Aeschylos "womanizes" the Persians in general by showing male lamenters, led by Xerxes himself.
But more commonly brought up in this context is the Capture of Miletos (473 or 472) by Phrynicus. This tragedy recounted the Persian destruction of that city in Asia Minor, which the Athenians were unable to prevent in time. It was therefore a raw reminder of Athens' failure in the early stages of the war. This particular play seems to have struck quite a nerve, and Phrynicus was fined for putting it on and upsetting the demos. Herodotos describes it:
Ἀθηναῖοι μὲν γὰρ δῆλον ἐποίησαν ὑπεραχθεσθέντες τῇ Μιλήτου ἁλώσι τῇ τε ἄλλῃ πολλαχῇ, καὶ δὴ καὶ ποιήσαντι Φρυνίχῳ δρᾶμα Μιλήτου ἅλωσιν καὶ διδάξαντι ἐς δάκρυά τε ἔπεσε τὸ θέητρον, καὶ ἐζημίωσάν μιν ὡς ἀναμνήσαντα οἰκήια κακὰ χιλίῃσι δραχμῇσι, καὶ ἐπέταξαν μηδένα χρᾶσθαι τούτῳ τῷ δράματι. (6.21)
"Now the Athenians, exceedingly traumatized by the sack of Miletos, made it clear in many ways, but especially when the poet Phrynikos created his tragedy "Sack of Miletos" and produced it, the whole theater fell into weeping, and the Athenians fined him 1000 drachmas because he had caused them to remember such unpleasantness so close to home, and they saw to it that the play would not ever again be put on."
This seems to have set a precedent that going forward, all Tragedy (that we can see in the evidence, at least) was not cleared to be centered in recent or current events, but instead in mythology, which could itself still be overtly related to current events. Aeschylos' Oresteia trilogy comes to mind in that regard, it is very much tied to Athens and Athenian identity. We don't know exactly how this played out, or why exactly. There was no expressed law or custom which forbade "historical" tragedy. It just seems to have faded away. Tragedy was inherently an agonistic activity, with competition and winners and losers, and it seems that mythology was just more fun for the audience and so naturally evolved further than historical tragedy. We will never know for sure.
Comedy remained hyper-current, and in fact seems to increase in its political overtones as the 5th century drew on. Perhaps the audience got their fill of current events and politics from Comedy and preferred to keep their Tragedy more abstract. Aristophanes plays like Knights and Wasps are political in nature to the extreme (both attacking Kleon, the popular demagogue of the day).
There is a great recent volume (actually two volumes!) on this topic by Matthew Wright, The Lost Plays of Greek Tragedy (Bloomsbury 2016-2019). Volume one is "Neglected Authors" and he talks about Phrynicus and many other interesting and little-known figures, and also the murky early history of Tragedy (Ch 1).