EDIT: In standard textbooks on classical mechanics I know after the notion of mass of a body is introduced, it is tacitly assumed that in all inertial frames the mass of a body is the same.
Does this fact follow from other basic principles of classical mechanics (like the Galileo principle of relativity) or it is an independent experimental fact?
A reference discussing this issue would be very helpful.
The idea from early treatments of special relativity that mass increases with velocity was superseded in general relativity and is better not used. It is a fundamental principle that the laws of physics are covariant - they are formulated using tensor (& vector & scalar invariant) quantities so as to be the same for all observers. Proper mass, or rest mass, is the invariant magnitude of the energy-momentum 4-vector $(E,\mathbf p)$ and satisfies (in units with $c=1$) $$m^2 = E^2 - \mathbf p^2. $$ There is no need for another concept of mass. There is no point in conflating energy with relativistic mass, since this only results from misapplying Newtonian equations instead of replacing them with relativistic tensor equations. We already have a good word, energy. There is no need to call it relativistic mass.
From a purely classical point of view:
"Does this [classical mass invariance] stem from other basic principles of
classical mechanics or is this an independent experimental fact?"
No, this fact doesn't stem from other classical principles. Classically, this is an independent experimental observation.
It boils down to Newton's 2nd Law for a classical object, ${\frac {{\it dp}}{{\it dt}}}=m \left( {\frac {{\it dv}}{{\it dt}}}\right) $. Here m is a constant ratio.
This is what Newton postulated, and what every classical experiment has since verified. This can't be proven using classical mechanics, because it is the foundation for classical mechanics.
This still holds true under a Galilean transform: Inertial reference frames do not affect any of the terms in Newton's second law. Experimentally, that constant ratio m still holds true - 'classical mass' is constant.
This is the explanation from the classical point of view. This is only an aproximation of reality at low energies. As the other answers show, the relativistic, four-space view is the "big picture", and doesn't need to seperate classical mass from invariant mass. A good reference regarding reference frames in classical and relativistic mechanics can be found here.
