How is $\sin 90° = 1$ possible? How can two angles of a triangle be equal to $90°$? If two angles were $90°$, this would mean that the two sides would be parallel and the angle of the third side would be equal to 0. Thus, there would be only two vertices and this wouldn't be a triangle at all, ultimately making $\sin 90° = 1$ impossible.

You can very well admit the concept of flat triangles (coming in two flavors, with a $0°$ and a $180°$ angle).
Anyway, the usual definition of the sine does not involve a triangle. Rather, the trigonometrical circle and the projection of a point at a given angle to the vertical axis. Indisputably, $\sin90°=1$. Simlarly, $\cos90°=0$.
