I find nice explanation by YakinAlan 171110976 "tane" is optional here. It provides some sort of a precision and emphasis. It implies that something has been counted already. For larger numbers like 100 or 1000 if you say, "Benim yüz kuşum var". I will (as a native ear) assume that that may not be an exact number, it is maybe something around that. Or even you are exaggerating or speaking off the top of your head. In that case, "tane" really helps. "Benim yüz tane kuşum var" may imply that you are serious about the quantity.
There are some places in which "tane" may be substituted with the item itself.
A: Benim üç kuşum var. B: Benim beş tane (var).
It is used for the questions: Such as "Kaç tane?"= "How many" etc.
I hope that helps...