Yes, it is true ( the irreducible constituents of the restriction of an irreducible character
to a normal subgroup all have equal degree). This is part of Clifford's theorem. It actually applies not just to complex irreducible characters or representations, but to irreducible representations over any field. In the case of complex characters, the uniqueness of the consituents of the restricted character follows because of the fact that the irreducible characters of $N$ form an orthonormal basis for the space of class functions of $N$ with respect to the usual inner product of class functions. For representations over other fields,
the Jordan Holder theorem can also be used.

Later edit: Clifford's theorem can be found in many texts: the module version states that if $S$ is a simple $FG$-module where $G$ is a finite group and $F$ is a field, and we have a normal subgroup $N$ of $G$, then ${\rm Res}^{G}_{N}(S)$ is semisimple, and all simple $FN$-summands are conjugate under the action of $G$ (so, in particular, all have the same $F$-dimension) and all occur with the same multiplicity as summands of the restricted module.