If $G$ is a compact connected Lie group, then the ring of characters is an integral domain (it is the ring of Weyl group invariants in the co-ordinate ring of a maximal torus). Let $r_1,r_2, R, T_1, T_2$ be as in the question. Assume that $r_1,r_2$ are ordinary irreducible representations
If $t_1,t_2$ are the dimensions of $T_1, T_2$, and $\chi _1$ and $\chi _2$ are the characters of $r_1,r_2$, and since the character of $R$ may be divided out in the integral domain, it follows from the assumptions that $t_1\chi _1=t_2\chi _2$. Hence $\chi _1$ occurs in the $t_2$ fold direct sum of $\chi _2$ with itself; that is, $\chi _1=\chi _2$. This is supposing that $r_2$ is an ordinary representation.
If $G$ is finite, then the comment of Richard Stanley already gives a counter-example.