Yes, see e.g. the paper "Arithmetically equivalent number fields of small degree" (Google for it) by Bosma and de Smit.
In brief: two number fields $K$ and $K'$ are said to be arithmetically equivalent if they have the same Dedekind zeta function. A famous group-theoretic construction of Sunada Perlis (Journal of Number Theory, 1977) gives many nontrivial (i.e., non-isomorphic) pairs of arithmetically equivalent number fields(remarkably, his . Remarkably, this construction applies works equally well to the construction of construct isospectral, non-isometric Riemannian manifolds)manifolds, as was later shown by Sunada.
Arithmetically equivalent number fields necessarily share many of the simplest invariants, for instance they have equal discriminants.
As the aformentioned paper explains, for arithmetically equivalent $K$ and $K'$, comparing zeta functions gives
$h(K)r(K) = h(K')r(K')$,
where $h$ is the class number and $r$ is the regulator. Therefore, to get an affirmative answer to your question you want a nontrivial pair of arithmetically equivalent number fields $K$ and $K'$ with $h(K) = h(K')$. The paper by Bosma and de Smit gives such examples.
