You are asking one of those "why" questions which really doesn't have a clear context. "why does 2+2=4?" I can give you various contexts for this matrix representation of translations which may satisfy your need for "why-ness" but the direct answer to any "why" question, since we're talking pure math here is "because the axioms so imply!".
First point, There actually is scaling but not of what you're paying attention to... Note that the matrix rep of translations does change the length of the representation vector (x,y) --> (x+ay,y). It does not map e.g. a unit vector to a unit vector which, for example a rotation does. We simply pay attention to the effect on a subspace (x --> x+ay) and choose our original vector (so y=1) to, by construction, represent translation this way. It is a matter of appreciating that --as you say-- shears translate whole planes onto themselves.
Second point, The translations form a continuous group (Lie group) and thus have a linear representations (in fact an infinite number of them). Any such group will have linear representations, its all a part of Lie theory. (linear representation = representation in the linear group = group of invertible matrices under multiplication).
Third point, [Which is probably more confusing than enlightening but let me give it a go...] You can embed the translation group in a natural way withing a larger (pseudo)orthogonal group. You can view the translations group as an extension of a rotation group, specifically with the group of rotations in d-dimensions, SO(d) = special orthogonal group in d dimension, you can extend this group to include the translations as well, ISO(d) = inhomogeneous special orthogonal group = the Euclidean group in d-dimensions. Both ISO(d) and SO(d+1) can be embedded in an indefinite case SO(d+1,1) which itself has a (d+2)x(d+2) matrix representation. They are subgroups leaving a given subspace invariant and thus will each have a d+1 dimensional representation if we choose the basis carefully and drop the invariant one. The inhomogenous case appears as a boundary to the homogeneous group's embedding (and there's an indefinite homogeneous case on the other side of this boundary.) To see the inhomogeneous boundary case you vary which 1-dim subspace you leave invariant and note the null vectors are on the boundary between "space-like" and "time-like" cases. The inhomogenous case is the subgroup leaving a null vector invariant.
In short there is a specific natural way to embed the inhomogeneous groups (which contain translations) within a higher dimensional homogeneous group (provided we allow indefinite metrics) which in turn has a fundamental matrix representation. This is to say translations are singular deformations of rotations. Geometrically we see this every day. We walk on Earth and imagine it is a translation but we are actually rotating about the Earth's center. The singularity is in the scale of our motion relative to the Earth's radius. The rotations expressed with trigonometric functions involve very small angles so we take the small angle limit and get translation. cos(x)=1, sin(x)=x = tan(x))
It manifests also in our imagining that boosting in space-time is simply a translation in the velocity space. It is in fact a pseudo-rotation expressible using hyperbolic trigonometry but the singularity is in our small distance scale relative to our time units (meters are a very small fraction of light-seconds) thus we can use the small pseudo-angle limit of the hypertrig. cosh(x)=1, sinh(x)=x=tanh(x).
You could create a specific boundary case, where you have a laser beam emitted from the north pole and you pick combinations of rotations about the center of the Earth and boosts (i think along the polar axis) such that the laser beam appears unchanged both in direction and frequency. That set of transformations would leave the event of t=0 at the center of the Earth invariant but the formal structure of that group would be the same as ISO(2) translations and rotations in some abstract plane. You would thus be expressing translations using 3x3 matrices (the 3-rotations and 3-boosts leaving t=0, r=0 invariant within the Lorentz group).
[I think I could further clarify this example by having the radius of the Earth move but... I have to get to work now.]
