How are the molar mass and molecular mass of any compound numerically the same?
So for example, the relative molecular mass of water is 18.02 u, and it's almost the same as the molar mass of water, which is 18.015 g/mol.
  
  So my question is, how is 18.02 u × (6.022 × 10^23) = 18.015 g/mol?


That is not quite correct.

The relative molecular mass $M_\mathrm r$ of water is a dimensionless quantity, i.e. the unit for relative molecular mass is the unit one and not $\mathrm u$:

$$M_\mathrm r=18.015\,28$$ 

It's the average mass per molecule $m_\mathrm f$ that has the unit $\mathrm u$:

$$m_\mathrm f=18.015\,28\ \mathrm u$$

Note that

$$1\ \mathrm{u}=1.660\,538\,921(73)\times10^{-27}\ \mathrm{kg}$$

Also note that you are missing the unit of the Avogadro constant:

$$N_\mathrm A=6.022\,140\,857(74)\times10^{23}\ \mathrm{mol^{-1}}$$

Thus

$$\begin{align}
M&amp;=m_\mathrm f\cdot N_\mathrm A\\[6pt]
&amp;=18.015\,28\ \mathrm u\times N_\mathrm A\\[6pt]
&amp;=18.015\,28\times1.660\,538\,921\times10^{-27}\ \mathrm{kg}\times N_\mathrm A\\[6pt]
&amp;=18.015\,28\times1.660\,538\,921\times10^{-27}\ \mathrm{kg}\times6.022\,140\,857\times10^{23}\ \mathrm{mol^{-1}}\\[6pt]
&amp;=0.018\,015\,28\ \mathrm{kg\ mol^{-1}}\\[6pt]
&amp;=18.015\,28\ \mathrm{g\ mol^{-1}}
\end{align}$$