Generally yes. But not always. Sometimes there's only a single such object. For example, there's a largest sporadic simple group. It is a very interesting object. But there's only one of it.

To use a slightly less silly example, up to isomorphism there's only one ordered complete archimedean field. We call it R and we care a lot about it.

Also, sometimes you lack enough data to know if there are other examples of what you care about. But yes, you should generally try to figure out if a non-trivial example exists before you start studying it.