Currently, tagging specific people to then search for them isn't possible on the Dropbox service.
However, there are two features that might help. The image search feature, which is available on certain plans, can find more generic items, like 'cat', 'tree' and so on. There is also the separate ability to tag any file as you like, so you could add people's names in that manner.
If you have any further queries, feel free to message back.