Examples of non-invariant yet "useful" properties of mathematical objects

An elliptic curve is said to be in Weierstrass normal form if it is defined by an equation of the form $$ y^2 = x^3 + ax + b $$ for constants $a,b$. This property is not preserved under automorphisms of elliptic curves: in fact every elliptic curve over a field of characteristic not 2 or 3 is isomorphic to an elliptic curve in Weierstrass form. Being in Weierstrass normal form is therefore not an intrinsic property of the elliptic curve itself, but of the presentation of the elliptic curve. The reason that Weierstrass normal forms are useful, despite not being preserved under isomorphism, is that it simplifies some calculations.

Something similar happens for matrices over $\mathbb C$. Every square matrix over $\mathbb C$ is similar to a square matrix in Jordan normal form. Clearly, the property of `being in Jordan normal form' is not preserved under similarities, which is the natural notion of isomorpism for square matrices in most situations. Again, the reason that we are studying the Jordan normal form (and similar decompositions) anyway, is that it helps when doing explicit calculations.

In setting up the categories of Banach spaces (or Hilbert spaces or inner product spaces or normed spaces), we choose to take continuous linear mappings as the morphisms rather than isometric linear mappings. The metric structure is not invariant under this choice for the morphisms, but the resulting categories are far more interesting and useful.