Question, because 鸡 is "chicken" and "蛋" is egg, I assume "鸡蛋" is a chicken egg...and then you use something else with 蛋 for other types of eggs, like “鸭蛋" = duck egg? or "鹌鹑蛋"=quail eggs? Is it not like in English where "egg" defaults to "chicken egg", because Chinese culture more widely consumes other kinds of eggs?

"Give me two chicken eggs" should work as well

給我兩個雞蛋。
You can also use 颗/顆(kē) for the classifier.

Isn't it possible to translate "两个" as "a couple"? I think I have seen this as a possible translation somewhere here on Duolingo, which made me wonder that.