amu and g/mol relation Do we have that $\pu{1 g/mol} = \pu{1 amu}$ ?

Because we have, for the mass of an atom of carbon 12, call it $m(\ce{^12C})$, that

$$m(\ce{^12C}) = \pu{12 amu}$$

and furthermore

$$\pu{1 mol} \cdot m(\ce{^12C}) = \pu{12 g}$$

therefore

$$m(\ce{^12C}) = \pu{12 amu} = \pu{12 g/mol}$$

So finally we get that $\pu{1 g/mol} = \pu{1 amu}$ .

However, my chemistry teacher is telling me that those are two completely different things and that I am confused between the mass per atom and the mass per $6.022\cdot10^{23}$ atoms. I can't understand how, and this is really bugging me, so help is very appreciated.

Note that this requires the mole to be a number (or a "constant"), which may be where I'm wrong.
You are correct, but to make it a little more clear you can include the assumed "atom" in the denominator of amu:

$$
\begin{align}
m_{\ce{C}^{12}} &amp;= \pu{12amu atom^-1} \\ \\
m_{\ce{C}^{12}} &amp;= \pu{12g mol^-1} \\ \\
\pu{12amu atom^-1} &amp;= \pu{12g mol^-1} \\ \\
\pu{1amu atom^-1} &amp;= \pu{1g mol^-1}
\end{align}
$$

In other words, the ratio of amu/atom is the same as the ratio of g/mol. The definitions of amu and moles were intentionally chosen to make that happen (I'm surprised your teacher didn't explain this, actually). This allows us to easily relate masses at the atomic scale to masses at the macroscopic scale.

To check this, look at the mass of an amu when converted to grams:

$\pu{1amu}= \pu{1.6605E-24 g}$

Now divide one gram by one mole:

$\pu{1g mol^-1}= \frac{\pu{1 g}}{\pu{6.022E23 atom}} = \pu{1.6605E-24 g atom^-1}$

It's the same number! Therefore:

$\pu{1g mol^-1}= \pu{ 1 amu atom^-1}$