It is always difficult to answer the question why did the authors phrase this in such and such a way, so I can only guess. My guess is that they assume certain background knowledge, which I now lay out.
The only input to the problem in the paper is $N$, which is given in binary, because, as the authors mention
If instead we were given $N$ in unary, there would only be one instance per problem size
It seems intuitive that if the problem were given in unary, then the problem would be $NP$-Complete instead of $NEXP$-Complete. This intuition is wrong, because then there would be a unary $NP$-Complete language. A result of Mahany  says that if unary $NP$-Complete languages exist, then $P=NP$ . Since the consequence $P=NP$ is believed unlikely, the unary version of the problem is probably not interesting, because nothing is believed to reduce to it. (In particular, therefore, it is of no use trying to reduce SAT to Unary-TILING)
They mention that
there would only be one instance per problem size, and the problem would trivially be in $P/poly$.
They might mention this for the same reason as above, namely, then there would be an $NP$-Complete language in $P/poly$, and then $NP\subset P/poly$, and a result of Karp and Lipton says that it follows that the polynomial hierarchy collapses (to $PH=\Sigma_2^P=\Pi_2^P$, and this result has subsequently been improved  to $NP\subset P/poly\implies PH=S_2^P$ and  to $NP\subset P/poly\implies MA=AM$). Perhaps the authors of your paper make these remarks because they assume that you are familiar with these results. If that was not the case, then you've learned something today! :)
 S. R. Mahaney. Sparse complete sets for NP: Solution of a conjecture by Berman and Hartmanis, Journal of Computer and System Sciences 25:130-143, 1982.
 R. M. Karp and R. J. Lipton. Turing machines that take advice, Enseign. Math. 28:191-201, 1982.
 J.-Y. Cai. S2P is contained in ZPPNP, Proceedings of IEEE FOCS'2001, pp. 620-629, 2001.
 V. Arvind, J. Köbler, U. Schöning, and R. Schuler. If NP has polynomial-size circuits, then MA=AM, Theoretical Computer Science 137, 1995.