The 2D Lorentz transformation can be derived from the following mathematical assumptions, which all have physical motivations.
It can be proved that there is a unique one parameter family of transformations ##L_v: \Bbb R^2\to \Bbb R^2##, defined for all ##v\in (-1,1)##, satisfying:
1. For each fixed ##(x,t)\in \Bbb R^2##, the mapping ##H:(-1,1)\to \Bbb R^2## given by ##H(v)=L_v(x,t)## is continuous.
2. Each ##L_v## (##v\in (-1,1)##) is a bijection, and its inverse is ##L_w## for some ##w\in (-1,1)##.
3. For each ##v\in(-1,1)##: Each line ##(t,x)=(t_0,x_0)+s(1,a)## (##s\in \Bbb R##), with ##a\in (-1,1)## and ##t_0,x_0\in\Bbb R##, is mapped by ##L_v## to a line ##(t',x')=(t'_0,x'_0)+r(1,b)## (##r\in \Bbb R##), for some ##b\in (-1,1)## and ##t'_0,x'_0\in\Bbb R##.
4. ##L_v(0,0)=(0,0)##, for all ##v\in (-1,1)##.
5. ##L_0## is the identity transformation on ##\Bbb R^2##.
6. For each ##v\in (-1,1)##: ##L_v(1,v)=(t',0)##, for some ##t'\in \Bbb R##.
7. For each ##v\in (-1,1)##: ##L_v(1,1)=(r,\pm r)## and ##L_v(1,-1)=(s,\pm s)## for some ##r,s\in \Bbb R##.
8. For each ##v\in (-1,1)##: If ##L_v(t,x)=(t',x')##, for some ##t,x,t',x'\in \Bbb R##, then either ##L_{-v}(t,-x)=(t',x') ## or ##L_{-v}(t,-x)=(t',-x')##.
Each ##L_v## (##v\in (-1,1) ##) is then given by ##L_v(t,x)=(1/\sqrt{1-v^2})(t-vx, -vt+x)## for all ##(t,x)\in\Bbb R^2##.
One needs not assume that ##L_v## is linear, for this follows, which was proved by micromass, strangerep and Fredrik in an in an old thread
https://www.physicsforums.com/threa...formations-are-the-only-ones-possible.651640/
for the general case when all lines are mapped onto lines, and it can be proved that it suffices to look at "timelike" lines.
Physical motivations:
1. It is possible to accelerate an object continuously to any speed less than light speed, through intertial frames.
2. The two frames are interchangeable. A consequence of the special principle of relativity.
3. An an object which is not being acted upon by a force, and hence moves with uniform rectilinear (timelike) motion, w.r.t one frame, is moving as freely w.r.t. the other frame. A consequence of the special principle of relativity.
4. Just an arbitrary practical convention about how we put marks on our rods and synchronize our clocks.
5. If ##v=0##, the frames coincide.
6. The relative velocity of Frame 2 w.r.t Frame 1 is ##v##.
7. A consequence of the invariance of the light speed.
8. This is about spatial isotropy. The transformation should still be valid if we change the directions of the spatial axes. See an earlier post by strangerep in this thread.
But it becomes more complex in 4D spacetime...