Is Nm the same unit of torque as mN?

Just like $2\times3=3\times2$, There is no difference between newton-meters and meter-newtons. They're two different ways of saying the same thing.

Probably your book is trying to avoid confusion when you learn about energy, which is also measured in newton-meters, although we normally rename the unit, when referring to energy, as joules.

You should go ahead and call the unit of torque "meter newtons" in your class, because that's what your instructor expects. But be prepared to see other people call it "newton meters". I'd even strongly recommend using newton-meters anywhere except in your class, since the unit ${\rm m\cdot N}$ (meter-newtons) is much too easily confused with $\rm mN$ (millinewtons).

The system of units has nothing to do with the physics behind the formulas, so it's pure convention. The SI is just a special system of units, which was made to be a standard, and maybe to make calculations easier (as there is simple transition rules between the different scales like m and km). (Also the SI was designed for the everyday life, I think, as the base units like m, s, etc. are well fit in in everyday acts like walking and talking. I.e. one's step length is around 1 m, and you can say a few words in 1 s.)

Technically mN (meter-newton) is not wrong, but a bit confusing for most physicists, as they will read that millinewton. Probably (one of) the most common notation(s) for torque is Nm, but foot pound is also commonly used, as mentioned in the comments.

In general there is lots of different notations for the same quantities, depending on the context. (1 angstrom = $10^{-10}$ m and 1 eV (electronvolts) $= 1.6\cdot10^{-19}$ J is used in atomic and molecular physics for example.) In case of torque for example, if somebody use a different unit for force (instead of N), than the units of torque will follow that convention, as it's an inherited/derived(?) quantity.

I think the Wiki page is mostly reliable about the (SI) system of units. (In general the English Wikipedia is reliable for studying, but obviously not for research.)

Edit: (One question about the radians was removed from the original post, but I leave the answer here. I hope it's not a problem.)

And about the radians... In SI it is considered as 'dimensionless' (as in SI base units its dimension is [m/m]=[1]), but in many cases it is straightforward to use rad to indicate that there is an angle in the equation. And also mrad (milliradian) is often used.

The unit for torque is $force \times distance$. So Newton-meters (N-m) is what it would be. However, since it is multiplication so meter-Newtons isn't really wrong, but no one says that.

So your teachers say Wiki isn't accurate. Okay. What did they say about every textbook out there? I'm curious if any of them have written at textbook and what they used. I find it amusing that your professor completely changed the subject by saying French textbooks were more reliable, thus weakly implying they would use mN, but did not actually say they did (and definitely showed no examples off his shelf). He basically changed the subject and dodged your entire question when he did that.