The connection of the "R" in the ideal gas law and osmotic pressure
It's not a coincidence at all!  If you do an online search for "derivation of osmotic pressure", you'll see how $R$ enters into the derivation.  

Indeed, that's one of the beauties of the van 't Hoff equation for osmotic pressure – it reveals that (under the modest simplifying assumptions of the van 't Hoff equation derivation) the osmotic pressure created by a dilute solute is equal to the pressure that solute would exert if it instead acted as an ideal gas!:

$$\text{van 't Hoff equation:  } \Pi_i = \frac{n_i}{V} R T = C_i R T$$

$$\text{ideal gas equation:   } p_i = \frac{n_i}{V} R T = C_i R T$$

Here the "$i$" refers to species $i$.