Is there a connection between M1V1=M2V2 and Boyle’s Law? Recently in Chemistry 101 we learned about $M_1V_1=M_2V_2$, where $M$ is the molarity of the two mixtures and $V$ is their respective volumes. 

The first thing that popped into my mind when I saw this formula is, “Hey, that reminds me of Boyle’s Law from high school physics - $P_1V_1=P_2V_2$ - the relationship between the volumes of a gas at different pressures.”

Is it just a coincidence that these two formulae look the same, or is there a connection between the pressure variable in Boyle’s Law and the molarity variable in the first equation? Are there other formulae which are structured similarly which are connected?
Yes, they are related. The first comes directly from the conservation of number of moles of the solute in a dilution,

$$n_1 = n_2 $$

Since $n = MV$,

$$M_1 V_1 = M_2 V_2$$

The second is related to the conservation of the total number of moles in an isothermal compression or expansion,

$$n_1 = n_2$$

Using the ideal gas law $n = pV/RT$,

$$\frac{P_1 V_1}{R T} = \frac{P_2 V_2}{R T}$$ 

Or $P_1 V_1 = P_2 V_2$.

Again, both are related to the conservation of moles (which by the way, isn't an universal law, since it's broken in trivial situations like the event of a chemical reaction), although the first one refers to solute only and the second to total number of moles.