Do 3 Boros Elite attacking trigger Battalion?
Yes.

Whenever a card refers to itself by name, it is referring to that specific instance of the card. Other cards with the same name are not affected or considered.

For a card to work in the way you describe, it would be worded significantly differently. Something along the lines of "Whenever Boros Elite and at least two other creatures not named Boros Elite...", or "Whenever Boros Elite and at least two other creatures with different names...". Examples where these restrictions are put in place are Escaped Shapeshifter and Gifts Ungiven