Why is linseed oil soap, and only linseed oil soap, sometimes recommended?

It is because linseed oil soap is superfatted, that is, it has an excess of fat and therefore has a treating effect on wood and other natural materials such as stone. It does not dry out but rather replenishes fat.

Västerbottenssåpa is made from rapeseed and corn and is also superfatted, that is, it works in the same way. A pine soap, on the other hand, is drying and is better suited in other contexts. Green soap is today usually colored pine soap and therefore does not have these caring properties either.
So this is the explanation for why linseed oil soap is sometimes recommended. It is the fat in the soap that is important!