The countable dense linear order is unique up to a non-unique isomorphism. The countable atomless Boolean algebra is unique up to a non-unique isomorphism. (Algebraically closed fields of a given characteristic and transcendence degree, and vector spaces of given dimension over a given field, are other examples already mentioned above.)
In general, if $T$ is any $\kappa$-categorical first-order theory (in a countable language), then the model $M$ of $T$ of cardinality $\kappa$ is unique up to a non-unique isomorphism. (This amounts to stating that $\operatorname{Aut}(M)$ is nontrivial. This holds e.g. because $M$ is saturated, and the number of its parameter-free $1$-types is strictly less than $\kappa$, hence there are two elements of $M$ with the same type, and one can find an automorphism mapping one to the other by homogeneity.)