Can I install a Eufy setup at a totally separate location and have a separate app on my phone? How would I go about doing that?
You don’t say what type of phone, but probably not. You can however manage multiple homebases in one Eufy app at multiple locations. Set up one homebase at each location and add them to your app.
A solution could be to create another account and then share the devices to your main account (having named the devices appropriately).