A Cayley graph or Caylay digraph \(\text{Cay}(G,S)\) is called a CI-graph of the group \(G\) if, for any \(T\subseteq G\), \(\text{Cay}(G,S)\cong \text{Cay}(G,T)\) iff \(S^\sigma= T\) for some \(\sigma\in \text{Aut}(G)\). The aim of the paper is to characterize finite abelian groups for which all minimal Cayley graphs and Cayley digraphs are CI-graphs. There exist infinite families of minimal Cayley graphs and Cayley digraphs of abelian groups which are not CI-graphs. Let \(G\) be an abelian group. (1) Assume that all minimal Cayley graphs of \(G\) are CI-graphs. Then either \(G\) is a 2-group, or \(G_2\neq H\times \mathbb{Z}_2\) for any \(H< G\) with \(\exp(H)\geq 4\). (2) All minimal Cayley digraphs of \(G\) are CI-graphs if and only if either \(G\) is a 2-group, or \(G_2\neq H\times \mathbb{Z}_2\) for any \(H< G\) with \(\exp(H)\geq 4\).