Is Bromide anion acting as a base in reaction with ethylamine and methylbromide?
The bromide anion is acting as a counter anion to balance out the ammonium charge in step 1.  

Given sufficient heat, step 2 can occur, with bromide acting as a weak base.  This is likely to produce gaseous produces which would drive the equilibrium as the gasses escape.  

Mechanistically, this is possible, but synthetically, it is more likely another base would be added in a typical synthesis.