Geometry is a part of mathematics. It starts with axioms, draws conclusions from them, and so on. "Alternative geometries" has nothing much to do with it being part of mathematics; people had used a set of axioms that they believed to be sufficient and it turned out not to be, and it turned out that adding any one of three possible additional axioms would produce one distinct mathematical geometry out of three possible ones.
On the other hand, practical geometry can be seen as either engineering or as science. There are many aspects that have nothing to do with mathematical geometry. For example, building precise instruments, how these instruments are affected by temperature, atmosphere pressure and humidity along a line that is measured, effects of the earth's curvature and of relativity.
For example, when you determine the location of points precisely using measurements in geometry, it all starts working fine just in agreement with mathematical geometry, but then you measure more precisely and the results disagree because (1) the earth is not flat but a ball, (2) because it is not a ball but a rotational ellipsoid, (3) because it is not a rotational ellipsoid but a flattened rotational ellipsoid, (4) because it is not a flattened rotational ellipsoid but a deformed flattened rotational ellipsoid. And then you take into account relativity...
So practical geometry is very much a science.