Why does the ideal gas law exactly match the van't Hoff law for osmotic pressure?

The law $PV = n RT$ gives the pressure $P$ of $n$ moles of ideal gas in volume $V$. Meanwhile, the law $\Pi V = n R T$ describes the osmotic pressure $\Pi$ due to $n$ moles of solute in volume $V$.

These are qualitatively very different situations, but there's a simple fundamental reason that they end up looking the same. Both of these laws are derived under the idealized assumption that the ideal gas/solute molecules don't interact with each other at all. So the expressions for the entropy of the ideal gas/solute are the same, and since the pressure of a system can be derived from the entropy, both situations yield the same pressure.

The reason that you see $\Pi V = n RT$ expressed in such different units in biology textbooks is simply because they're using the units that are most convenient for them.

I'm answering your 2nd question:

It's a really easy proof :

You have $C_m=\frac{m}V$ and $C=\frac{n}V$ where $n=\frac{m}M$, thus $C=\frac{C_m}M$

Therefore : $$\Pi=\frac{C_m R T}{M}$$ For $N$ the ions number it's related to biology, the ions are interacting and they cause this pressure, For example if it's $\mathrm{NaCl}$ the ions are $N=2$

Thus Van't hoff law is written in Biology and Biophysics textbooks : $$\Pi =\frac{NC_mRT}M$$