In addition to what has been said already: I think that everywhere in Mathematics when you speak of symmetries you mean "group plus action" and not just the group itself.


The thoughts about symmetry are probably of geometric nature: asking for symmetry means asking for the symmetry of a geometric object (we have already the examples of Riemannian manifolds, but there are many more) In differential geometry you can ask for "symmetries" of all kind of structures: metric, but also symplectic forms or Poisson tensors. In this case you enter the realm of dynamical systems with symmetries. The symmetries usually help to simplify the dynamical system by using "conserved quantities" to eliminate degrees of freedom. You may remember this from your first mechanics courses when dealing with the Kepler problem...

But symmetries in crystals might yet give another example, not related to Lie groups and some inherited action from a linear action: treating a crystal as an abstract lattice with colored edges and vertices one may well ask for its symmetries and arises at discret groups acting in a much more combinatorial way. The original possibility that the lattice can be embedded into some Euklidean space is no longer relevant.


In addition, symmetries arise in much more abstract concepts that these geometric ones. A prominent example is perhaps the question of solving polynomial equations. Here the symmetries of the polynomial might allow for general formulas or not. This is the beginning of Galois theory in field theory, where not Lie groups but discrete groups are acting.


From my own field a statement which I would like to understand better: the Grothendieck-Teichmueller group acts on the set od Drinfeld associators. Not a linear action at all :(


On the other hand:
One reason why linear actions are so omnipresent is perhaps that (beside being the simplest type of actions) all types of geometric actions dualize to a linear action on the spaces of reasonable functions on the geometric spaces. Hence even a group action on some geometric object (manifold, lattice, ...) can by studied by means of representation theory when one looks for the induced action (via pull-back) on the functions on it. However, this is typically quite complicated as the representation spaces typically are infinite-dimensional.