Here is a fact that I find interesting. Strictly speaking it is to say that a certain number counts three completely unrelated things, but it seems easy to make these into a combinatorial problem or to define an explicit bijection between two (or three) sets.
So, the concrete example is $168$, which
(1) the number of hours in a week;
(2) the number of primes under $1000$; and
(3) the size of the smallest simple group of Lie type.
$168$ is also $4 \times 42$, where $42$ is that famous number Douglas Adams wrote about. (How did he know?) Which is of course the reciprocal of the smallest positive number that can be written as $$1-\frac 1a -\frac 1b -\frac 1c$$ with $a,b,c\in \mathbb N$ and coincidentally (and relatedly) the largest size of the automorphism group of $C$ a smooth projective curve of genus at least $2$ is $42\cdot {\rm deg} K_C$ and of $S$ a smooth projective surface of general type is $42\cdot K_S^2$(42 K_S)^2$.
So I guess one could add that $168$ is also
(4) the maximum value of $\frac 4{1 -\frac 1a -\frac 1b -\frac 1c}$ with $a,b,c\in \mathbb N$; and
(5) the largest possible size of the automorphism group of a smooth complex projective curve of genus $3$,
but I admit the last two are a little artificial...
