A four-vector is defined to be a four component quantity $A^\nu$ which transforms under a Lorentz transformation as $A^{\mu'} = L_\nu^{\mu'} A^\nu$, where $L_\nu^{\mu'}$ is the Lorentz transformation matrix, which including boosts, rotations and compositions. (In other words, as the components of a position vector $(x0,x1,x2,x3)$would transform).
The useful property of four-vectors is claimed to be that if two four-vector expressions are equal in one frame, then they will be equal in all frames :
$A^\mu = B^\mu \Leftrightarrow A^{\mu'} = B^{\mu'}$
and therefore we can express laws of physics in terms of four vectors, because they remain invariant in all frames.
But this property will be true even for four component quantities that transform (across reference frames) as $A^{\mu'} = T_\nu^{\mu'} A^\nu$, where $T$ is any transformation matrix (not necessarily a Lorentz one). As long as we can find a $T$ that will describe how the quantity's components transform, we can apply that T to both sides of an equality.
So why require (i.e. define) four-vectors to only be quantities that transform under a Lorentz matrix?
Best Answer
Ultimately, you make a totally legitimate point; we very well could have defined the term "four-vector" to refer to a type of object that transforms in a different way, but we make the particular definition that we do because it is useful to have a term that refers things that transform like spacetime positions when you change frame. Here are two reasons why:
Fact 1. Given any two four vectors $A^\mu$ and $B^\nu$, the quantity $g_{\mu\nu}A^\mu B^\nu$ is invariant under a change of frame.
Notice that this would not have been true unless $A^\mu$ and $B^\nu$ were four-vectors because the proof of this fact relies on the metric being preserved by Lorentz transformations, and not by other arbitrary things. Here is another reason why the definition is useful
Fact 2. Lots of really useful and physically significant quantities happen to be four-vectors. Take, for example, $J^\mu$ and $A^\mu$ (the current and vector potential) in electromagnetism.
Having said all of this, however, note that there are tons of other quantities that do not transform as four-vectors when one changes frame. In fact, given any representation $\rho$ of the Lorentz group, one often encounters quantities $Q$ that transform as $$ Q' = \rho(\Lambda) Q $$ For example, there are objects called Weyl spinors that transform as $$ \psi' = \rho_\mathrm{weyl}(\Lambda)\psi $$ when one transforms between frames.
The upshot of all of this is the following
Upshot. Lorentz 4-vectors are not special. However, since every change of reference frame can be associated with a Lorentz transformation, every quantity that you want to transform between frames must necessarily transform in a way the depends, in some way or another, on the Lorentz transformation between the frames. This leads us to not only define four-vectors, but a host of other objects that have specified transformation laws under changes of frame and to give them special names. Doing this is useful because such objects appear all over the place in physics, and we can prove useful properties about objects with certain transformation behaviors.