While one would think that this is only important for things you're doing that you don't want the government to know about (see [1] page 52 for details on how not to mess this up -- basically don't have them turned on together, don't turn one off and turn the other on in the same place, or log in to the same sites or store the same numbers on both phones), it's also important for Facebook and other private tracking. If you have Facebook on your burner phone and your friends have Facebook on their phones with location enabled, it's over [2].
In the US, different government departments define homelessness different ways [1]. How and why they came to do that would be a very interesting read. I had looked into this to see what it would take to legally check the "I'm homeless" box on the ACA exemption form, which I believe was under the HHS rule you posted. But that's different from what it would take to get housing assistance from HUD, and that's probably even different than how the average person would define "homeless".
[1] https://www.defcon.org/images/defcon-22/dc-22-presentations/...
[2] https://splinternews.com/facebook-is-using-your-phones-locat...