How can normalities be interchangeably used?
The fundamental idea behind gram equivalents and normality is that one gram equivalent of an acid reacts with one gram equivalent of a base. This idea eliminates the need of using molarity and thinking about mole ratios for any acid base titration.

Equal volume 1 N HCl will react with 1 N NaOH. Similarly, 1 liter of 1 N $\ce{H2SO4}$  or 1 N $\ce{H3PO4}$ will consume 1 L of 1 N NaOH.

The key calculations is that you should know how to calculate normality from a given molecular weight and a balanced equation:

For example: 
$\ce{HCl}$ -> $\ce{H+}$ + $\ce{Cl-}$
gram equivalent weight= formula weight/ (No. of acidic protons) = 36/1 =36

$\ce{H2SO4}$ -> $\ce{2H+}$ + $\ce{SO4^2-}$
gram equivalent weight= formula weight/ (No. of acidic protons) = 98/2 = 49

Can you now make the connection that why "The part 1.80 N HCl can also be written as 0.9 N H2SO4" is quite wrong? because

1 N HCl = 1 N $\ce{H2SO4}$

and both solutions contain 1 gram equivalents of protons.

Yes if your teacher were comparing normality and molarity, then it is different.

Molarity calculations are different and far more easier. By definition, M of X= moles of A/ Total volume in which X is present.

A 1 M HCl just means there is one mole HCl (gas) dissolved in 1 L of solution
Similarly, 
1 M $\ce{H2SO4}$ is 1 mole $\ce{H2SO4}$ dissolved in 1 L of solution.

If you were to titrate  1 L of 1 M NaOH with 1 M HCl you would require 1 L of HCl. However, since  sulfuric acid furnishes two moles of proton $\ce{H+}$ for each mole of sulfuric acid, only 500 mL would be required.

In short, 1 M HCl is equivalent to 0.5 M $\ce{H2SO4}$ in terms of titer for $\ce{OH-}$ ion.

Contrast this molarity unit with normality which is almost obsolete except in some South Asian colleges.