Yes, it's electrophylic aromatic substitution but, as agrobert said, NEt4Br cannot play the role of FeCl3, since it's not a Lewis acid.
The role of NEt4Br is to form, by reaction with Br2, NEt4Br3, which is a well known brominating agent. This reagent can be isolated in form of orange crystals, and is much easier to handle than liquid Br2 (though in this case, since it's prepared in situ, liquid bromine is used anyway).
I think the reason to use it in this example is to obtain a good regioselectivity. In the molecule of the example, the selectivity for EAS is in ortho and para positions (refered to the amine) and the use of a Lewis acid such as FeCl3 would direct the reaction preferably to the ortho positions, due to complexation of the Fe with the nitrogen.
In the other side, NEt4Br3 is known to have selectivity to the para position. Also, it's a mild brominating agent, so it just forms the monobrominated product.