It's just not true that having isomorphic Lie algebras implies a bijection between the irreducibles (presumably you mean a bijection compatible with the isomorphism between the Lie algebras). For example when $G = SU(2), H = SO(3)$ only half of the irreducibles of $SU(2)$ come from irreducibles of $SO(3)$.
Continuing, the natural double cover $SU(2) \to SO(3)$ identifies the algebra $C(SO(3))$ of continuous functions on $SO(3)$ with a subalgebra of $C(SU(2))$ (exactly the subalgebra of "even functions" with respect to $-1 \in SU(2)$) and in the Peter-Weyl decomposition this subalgebra corresponds to the irreducibles of $SU(2)$ which descend to irreducibles of $SO(3)$ ("integer spin"), while there's a whole other subspace of odd functions corresponding to the irreducibles which don't ("half-integer spin").