At the end of writing up our manuscript, we discover that there has been another earlier study in a very different context (but using a physical model very similar to ours) that claims a similar result.

We were not inspired by this earlier work and we could not have found it until we had our results; so this is not a case of not having done proper literature survey a priori.

So, do we cite them within the main text despite not having been inspired by it at all? Or do we cite them at the end highlighting the similarities?

In either case, our results certainly complement theirs.

At the end of writing up our manuscript, we discover that there has been another earlier study in a very different context (but using a physical model very similar to ours) that claims a similar result.

We were not inspired by this earlier work and we could not have found it until we had our results; so this is not a case of not having done proper literature survey a priori.

So, do we cite them within the main text's introduction and results despite not having been inspired by it at all? Or do we cite them during the concluding remarks highlighting the similarities?

In either case, our results certainly complement theirs.