There's a difference between hardware support (new battery, replacement screen, etc) and software support.
Typically, Apple provides hardware support for 5 years after manufacture on their products.
The iPhone 5S is not currently listed as 'vintage'. However, it's possible the list may be updated shortly, as the phone was discontinued 5 years ago.
The iPhone 6 was discontinued 2 years ago, so will still receive hardware support until 2022.
As for software support: obviously, the devices will not be offered new OS updates, but in the event of a major security flaw, they may receive a patch to fix the problem.
The phones will of course continue to work as they are. Apps will not be updated if they require newer OS versions than are present.