An affine transform is a linear transform plus displacement (linear transforms by themselves don’t move the origin).
An affine transform in three dimensions has much more degrees of freedom than you described. Generally 12 – the 9 elements of a 3x3 matrix of the linear transform plus the 3 elements of the displacement.
What you mentioned is an orthogonal linear transform plus displacement.
This is the class of transformations that only ‘rotate’ (do not change length of vectors, do not change angles between vectors). Together with the displacement this is the class of ‘rigib body’ transforms. Things that don’t break apart solid bodies …
It has 6 degrees of freedom (3 for the rotation, 3 for the displacement).
Given an action on one point (where it was before the transform, and where it is afterwards), you are given only 3 'knowns" (not 6 as you might think…)
Even given two actions is not enough (even though you get 6 “knowns”). You will still miss one parameter.

Affine transformations have the property of
keeping parallel lines parallel after transformation. You can think of this as the
rigid-body transformations plus scaling and shearing.