The fact that the angles of a figure are congruent does not automatically mean that the sides are.
An equiangular (but not regular) hexagon:
Two equiangular octagons, one regular, one not:
In the case of triangles, being equiangular requires that the triangle also be equilateral. That is, every equiangular triangle is a regular triangle. However, this is not the case for all polygons. For example, a rectangle is equiangular -- all four angles are 90° -- but need not be square (need not have all four sides the same length). Thus, not all equiangular quadrilaterals are equilateral and so are not all regular.
The prefix equi- is from Latin and means 'equal'.