It is a combination of phone and network. However there is a carrier-specific configuration that is pushed onto the phone and activated depending on which SIM card is inserted. You can see this in settings->General->About, look for "carrier" and notice a name and version number in that field.
So when you have your Verizon SIM card in there, the phone gets configured to force tethering traffic over a separate APN to which Verizon can attach special behaviors. T-Mobile does the same thing but the special behaviors are less onerous. Other carriers have their own policies reflected in their own specific iPhone carrier settings.
This is the same mechanism by which an LTE connection shows as "LTE" when I have my T-Mobile USA SIM card in the phone, but "4G" when I have my DTAC one in there.