A system $\mathcal{T}$ is linear if its response to a weighted sum of two signals equals the weighted sum of its individual responses to those two input signals:
$$\mathcal{T}\big\{\alpha x_1+\beta x_2\big\}=\alpha\mathcal{T}\big\{x_1\big\}+\beta\mathcal{T}\big\{x_2\big\}\tag{1}$$
with arbitrary constants $\alpha$ and $\beta$, and arbitrary input sequences $x_1[n]$ and $x_2[n]$. A system $\mathcal{T}$ satisfying $(1)$ is completely characterized by its impulse response $h[n]$, and its input-output relation can be formulated as a convolution of the input sequence with its impulse response:
$$\mathcal{T}\big\{x\big\}=\sum_{k=-\infty}^{\infty}h[k]x[n-k]\tag{2}$$
It is easily shown that the given system
$$\mathcal{T}\big\{x\big\}=y[n]=ax[n]+bx[n-3]\tag{3}$$
satisfies $(1)$, and that it is characterized by the impulse response
$$h[n]=a\delta[n]+b\delta[n-3]\tag{4}$$
Clearly, its input-output relation $(3)$ can be written as a convolution sum $(2)$. Equivalently, the $\mathcal{Z}$-transform of its response is given by the multiplication of the $\mathcal{Z}$-transform of its input sequence and its transfer function $H(z)=\mathcal{Z}\{h[n]\}$:
$$Y(z)=X(z)H(z)\tag{5}$$
By contrast, an affine system does not satisfy $(1)$, and it cannot be characterized by an impulse response or, equivalently, by a transfer function. The given linear system $(3)$ could be made affine by adding a constant $c$ ($c\neq 0$) to its output:
$$\mathcal{T}\big\{x\big\}=y[n]=ax[n]+bx[n-3]+c\tag{6}$$
This input-output relation cannot be formulated in terms of a convolution $(2)$, and it can easily be checked that the system $(6)$ doesn't satisfy $(1)$. Such a system is not linear, since part of the output (the constant $c$) does not depend on the input signal $x[n]$.
There is no reasonable definition of linearity according to which the given system $(3)$ could be classified as being non-linear.