The answer is no.
It is true of triangles, but not necessarily of other polygons.
A good counterexample is the Rhombus.
You can define a Rhombus as a quadrilateral with 4 congruent ( equal) sides.
However, only the opposite angles are equiangular, not all 4 angles.
Picture, if you will, a very elongated Rhombus to easily see this.
The only equiangular Rhombus is the Square.
Many regular polygons are equiangular but not all.
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A regular polygon has equal angles AND sides.
Not only can a regular polygon have equal angles and sides, it must have them.
A polygon with sides of equal length and equal angles is termed a regular polygon.
A polygon with four equal angles and sides is a square.
A polygon has 3 or more sides and is called regular when all sides and angles are equal and it is irregular when its sides and angles are not equal.