Can anybody give me an example of a "naturally-occurring" algebraic category $C$ in which:

1. $C$ has two non-isomorphic objects $A$ and $B$ which are bi-embeddable via monic maps; but

2. $C$ does NOT have any infinite collection $A_{1}$, $A_{2}$, ... of objects which are pairwise bi-embeddable (via monic maps) and pairwise nonisomorphic?

Alternatively: can anybody give a reason why, under some reasonable hypothesis about the category $C$, property 1 should imply that 2 fails ("there is an infinite collection of pairwise bi-embeddable, pairwise nonisomorphic objects")?