Never used the word circle in my definition of pi. If planes curve in another geometry then no. I think these would called surfaces rather than planes. You could define a plane as simply the set of points sharing a common normal vector not intersected more than once by lines oriented in directions that have a non-zero normal component (i.e. parallel planes are unique).

In hyperbolic and elliptical geometry ratio of radius and circumference of a circle is not a constant. You can't define pi there as such.

Anyway, pi comes up in various fields of math with no relation to the geometry of space we live in.

Ironically, pi comes from geometry (land measurement), and the land they measured has elliptical geometry.