Not sure if this is really how the card is supposed to work, but Born Predators says 'deal X dmg to target unit where X is no. of lizardmen cards you control'. Does it count the supports like the shrine and cloak which have the word Lizardmen in their trait list as well?

Yep. Skaven cards were similar with their abilities. Simply counting the number of times you see the "lizardmen" trait on your side of the table.