It's a completely standard Cartesian projection (x = longitude, y = latitude), but with nonsense graticules plotted on top. I think BBC took a NASA image of Mars and added their own graticules and landing-site labels. Compare the BBC image with these two NASA images:

* BBC: https://ichef.bbci.co.uk/news/976/cpsprodpb/825B/production/...

* NASA base map of Mars: https://astrobiology.nasa.gov/uploads/filer_public_thumbnail...

* NASA landing sites: https://mars.nasa.gov/resources/24729/map-of-nasas-mars-land...

It basically looks like equirectangular and modified with pointy poles.

https://en.wikipedia.org/wiki/Equirectangular_projection

Maybe it is just regular equirectangular but the lines are drawn weird.

It looks like the equirectangular Mars projection. Details can be found on the HiRISE or THEMIS sites.