counter for long, thin things (e.g. swords, pens)
Used specifically for long, slender objects such as swords, pens, pencils, and sometimes branches. Not a general counter for all thin items.
This katana is one fine sword—no, one fine blade.
Please give me three pencils.
本 is the general counter for long, cylindrical objects (bottles, trees, etc.), while 枝 is more specialized for slender, often bladed or stick-like items.
振り is a counter specifically for swords, emphasizing the weapon aspect, whereas 枝 can also count other long, thin objects.
The counter 枝 originally derives from the noun 枝 (えだ, 'branch'), extended metaphorically to count long, slender objects.