So I understand that homogeneous coordinates allow us to encode n - 1 dimensions with n dimensions since they are all equal up to a scalar. That means the vector (2A, 2B, 2) should encode the same translation in 2D as (A, B, 1). i.e,
$\begin{pmatrix}
x \\
y \\
1
\end{pmatrix}$$\begin{pmatrix}
1 & 0 & A\\
0 & 1 & B \\
0 & 0 & 1
\end{pmatrix} = $
$\begin{pmatrix}
x \\
y \\
1
\end{pmatrix}$
$\begin{pmatrix}
1 & 0 & 2A\\
0 & 1 & 2B \\
0 & 0 & 2
\end{pmatrix}$
Except they don't, since the LHS is equal to $\begin{pmatrix}
x + A \\
y + B \\
1
\end{pmatrix}$ and RHS = $\begin{pmatrix}
x + 2A\\
y + 2B\\
2
\end{pmatrix}$
These two vectors are not proportional.
So where am I going wrong? Where does the 'homogeneity' of these coordinates come into play with affine transformations?
Short answer: you need to scale the whole matrix, i.e. $$\begin{pmatrix}2&0&2A\\0&2&2B\\0&0&2\end{pmatrix}$$
Homogeneity is a property that works well will multiplications, but not so well with additions. If scalar multiples are irrelevant, in a purely multiplicative expression such a scalar factor will merely propagate to the result without changing geometric meaning at any point. If you have additions, however, the result tends to depend on the specific representatives chosen for the input.
One way to describe the matrix you had would be as a sum of the first two columns of the unit matrix and a matrix that has the displacement embedded into its third column. That way the general problem observed with additive expressions applies to that formulation as well. Making all matrix entries scale together ensures that a scalar factor in the input propagates fully to the result.