When are two Abelian p-groups with the same character isomorphic?

Question

Definition 2.6 of this journal paper defines a notion called “character” for Abelian $p$-groups:

1. $\oplus_\alpha H$ denotes the direct sum of $\alpha$ copies of $H$ where $\alpha\leq\omega$.
2. If $A=\oplus_{i<\omega}\mathbb(Z)(p^{n_i})$, then the character of $A$ is $\chi(A)=\left\{\langle n,k\rangle:card(\left\{i:n_i=n\right\})\geq k\right\}$.
3. If $G = A\oplus\oplus_\alpha\mathbb{Z}(p^\infty)$ for some $\alpha\leq\omega$ and some $A$ as in part 2, then $\chi(G)=\chi(A)$

Where $\mathbb{Z}(p^n)$ denotes the cyclic group of $p^n$ and $\mathbb{Z}(p^{\infty})$ denotes the Prüfer $p$-group.

Now suppose you have proven that two Abelian $p$-groups $G$ and $H$ have the same character. Then my question is, does that get you any closer to proving that $G$ and $H$ are isomorphic? If so, what do you need to prove in addition to that to prove that they’re isomorphic?

Does the fact that they have the same character restrict how much their Ulm sequences can differ?