This looks like Trileçe which is indeed a Turkish take on tres leches:
Trileçe’s heritage is a lot more complicated than that of the éclair. Chasing down the elusive origins of this cake – a Balkan cousin of the Latin American classic tres leches – leads to a deep, dark, global rabbit hole.
And it's certainly got milk in it!
At Köfteci Arnavut they claim to mix kaymak, the heavy cream of water buffalo milk, and cow’s milk together for the characteristic soak given to the cake. At Baltepe Pastanesi, İdris Beğiroğlu called his recipe a secret but admitted that there was plenty of krem şanti, whipping cream, in there. Tuğra’s menu lists it as a cake of three milks – cow’s, goat’s and sheep’s.