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The problem comes from using a symbolic variable.

This means f and g as defined in your code live in Sage's SymbolicRing.

To work with polynomials, better work in the polynomial ring you define.

Achieve that by defining x as the generator of the polynomial ring R.

from sage.all import *
R = ZZ['x']
x = R.gen()
f = 3*x**2 - 3*x**8
g = x**2 - 1
print g.degree(x)
h = f.mod(x**2 - 1)
print h.degree(x)
print f.quo_rem(g)

Now x is the generator of R and everything works as expected.