Is it possible to be enrolled at a Masters program at one university and a PhD program in another university?
Is one allowed to do this? Or is this not possible in general?
It is possible. Many PhD programs don't require a masters, just an undergrad, so it is definitely possible, but as mentioned, why would you want to do that? Is it in complementary subjects that you plan to you later on? It would be awfully hard to do well in both programs. They are much more demanding than undergrad. I would suggest first doing the masters and then moving to the PhD, but it is your choice.