In this context "support for the war" means, in a world where there are several awful conflicts going on at any given time, supporting active intervention in a particular conflict by giving support to one side.

For example, I support the west providing weapons to Ukraine - but I would not support a US return to Afghanistan to overthrow the Taliban, or an intervention in Mexico to wipe out drug cartels, or an invasion of China because of the internment of Uyghurs - even though those are all awful.

Politicians will often go as far as explicitly announcing the criteria by which they'll decide which wars to involve their country in - for example https://en.wikipedia.org/wiki/Weinberger_Doctrine