There are underlying similarities between Turkish and Greek music at a deeper level, and there are clear similarities between Greek pop and Turkish pop at a more proximate level.
At a deeper level, the scales and instruments used by Turks and Greeks are related, through close to a millennium of coexistence. The tunings and modes of Byzantine chant have undergone microtonal influence from Turkish classical music. (The Greek chant preserved in Corsica does not have the same microtones.) Both the wind and the stringed instruments of the region have travelled in both directions. And of course the folk music of Christians and Muslims living in the same region was usually indistinguishable, whether it was Greece or Turkey. (Crete appears to have been an exception, but Christians took up the Muslim repertoire anyway after the Muslims left.)
At a more proximate level, the pop tradition of Greek music started with Rebetiko, and Rebetiko itself is clearly rooted in Smyrneika (Ottoman café music associated with Smyrna/İzmir). The scales and style were clearly Ottoman in the 1920s, although they evolved in more Western directions in Greece, from the 1930s (the Piraeus style) on.
Even before Rebetiko, there was a parallel outright Western pop tradition in Greek music, and the two traditions have mostly remained distinct; rebetiko has turned into Laïko (which in other answers I’ve termed “bouzouki pop”), and Western operettas of the 1910s have given way to rock, R&B stylings, and Euro disco.
Greeks are well aware that their Laiko tradition, and the revered Rebetiko that it comes from, has Turkish roots. Occasionally a nationalist might grouse about, but for the most part they’re happy that they’ve nativised it. It is worth noting that the Rebetiko that there is the most reverence for is not the Smyrneika of the 1920s, but the more nativised Peiraeus style of the 1930s.
With the thaw in Greek–Turkish relations since the 1990s, there has been a lot of traffic of “serious” musicians between the two countries; the Wikipedia article I linked to points out that contemporary performers of Rebetiko, if anything, overemphasise the Turkish style of the music. There has been some traffic of pop songs between the two countries, although that in itself is not remarkable; Arabic and Indian pop songs have also been covered in Greek.
In discussion with Turks on Quora, I’ve found that we understand each others’ music, it is familiar to us—but also that we would not mistake one’s music for the other. One user (and I’m annoyed I don’t remember who: it was in a comment, so good luck searching it) offered to me that there was something more impassioned about Turkish music, and more fatalistic about Greek music (“What can you do? Let’s have an ouzo”). That threw me, but then I realised that what came across to them as fatalistic comes across to me as stern and restrained: that was the contribution of the Piraeus style.