Bringing ESAN-type dogs onboard BA flights used to be possible for a brief time in the late 2010s when BA was forced by US ACAA regulations (Air Carrier Access Act) to allow them on direct flights to and from the US only, and very reluctantly did comply with those regulations. Look up old versions of the 'traveling with animals' section on the BA website ca. 2018 to 2020.
The ESA thing ended rather abruptly in January 2021 following regulatory changes in the US that allowed BA (and all other airlines) to close its cabins for them again. Those instagram posts surely dated back to that time, and I`m pretty sure the show dog above got into the cabin under these rules as well. Back then, airlines had no regulatory backing questioning the bought-off-the-internet service dog vests that people used far and wide to get their animals into aircraft cabins for free.
All that is over now, though. Since January 2021, the only dogs you see in BA cabins are bona-fide, fully-vetted-by UK-sanctioning-bodies, 'real' service animals.