The Apple AirTag, when paired with an iPhone, are actually associated to your Apple ID, not the phone itself. In theory, any other iPhone where you are logged in with the same ID, should be able to "see" the tag in the Find My app.
In addition, the AirTags rely on both Bluetooth and the Apple Find My network (which I'm sure you know consists of literally millions of other iPhones) to get the tag's "location" updates ... actually they would be iPhone GPS location updates when that phone is within Bluetooth range of the tag.
Ok, so in this case, leaving your other phone at home really won't make any difference. Should your dog leave the Bluetooth range of either of your phones, it will require other iPhones to pick up the tag's ID. If successful, it will relay the tag's ID, along with that phone's GPS location to the Apple servers over an Internet connection. That's where you would see any location updates with the Find My app on the phone you are traveling with.
It's important to understand that the AirTag was not designed for tracking, either pets or people, but primarily designed to help you locate a missing stationary object, like car keys. If knowing your dog's location is critical to you, you should consider getting a dedicated GPS tracker for each of your dogs.
Going back to your iPhone. It has to be a 6s model to work with the AirTags. An iPhone 6 will not work.