An amphibious flagship can be a navy's capital ship

Do modern LHDs and LPDs count as capital or prestige ships?

For many navies, yes: they are flagship-scale assets whose prestige rests on unusually broad operational capability.

Every amphibious ship under discussion can count as a capital ship for its own navy. Western examples tend to be larger because they are expected to travel farther and operate in demanding waters such as the North and South Atlantic. LHDs and LPDs also carry prestige, but that status rests on real lift, command, aviation, and humanitarian capacity. Algeria's flagship and France's preference for several Mistral-class ships alongside one carrier show how amphibious hulls can anchor national naval power without being battleships or fleet carriers.