Am I right to say in an Exchange Economy with two consumers with identical preferences, the equilibrium price (WEA) of the two goods would be determined by availability of two goods, i.e, total endowment? Whereas if their preferences are not identical, this is not true, the equilibrium(WEA) would be determined by utility functions and endowments together?
No you are not. Preferences determine the equilibrium, even if they are identical, because they determine the value of the endowments.
Consider two agents, one ("A") having ice cream only, and one ("B") having lava only.
Case 1: Both hate lava
Assume lava burns tongues and is useless (typical Economist). Then, the initial endowments of B have zero value, as no one wants them. The equilibrium allocation equals the initial allocation.
Case 2: Both find lava useful
Now, imagine lava being useful for lava lamps (whatever). They still have identical preferences, but now they attach some positive value to lava. The value of B's initial endowment has changed, and hence he can trade at least some of it for ice cream. The equilibrium allocation no longer equals the initial allocation.