That’s no coincidence: the Israeli military’s upper ranks are dominated by ex-special operations types, and their influence tends to trickle down to the troops.
Meanwhile, in the US Army dress uniform (at least until the recent change, I haven’t seen the new regs) you **could **have distinctive unit crests (pin-ons, for your breast pocket or epaulette) for some “regimental”-equivalent level units e.g. 130th Engineers, 292d Support Bn, 162d Field Artillery, 296th Infantry. (Nowadays in many branches “regiment” has come to mean batallions that for ceremonial/historic purposes maintain continuity of the regimental identity. As in, formally they are designated “1st” Bn of the 296th Infantry, but really they’re just the one batallion and they operate as part of the 92d Brigade.)
However on both the dress and combat uniform jackets, they will wear on the sleeve as indicated previosuly the patch of the operational command formation to which that unit reports, at brigade-level or above (thus for instance the last two units I mention would wear the patch of the 92d Infantry Brigade).
Still wear pin on insignia, even in ACUs, which is a great point. The Distinctive Unit Insignia, also called a unit crest, is worn on the beret flash and will denote the unit’s regimental identity as described above. The flash itself, may even be unique to the battalion level. For instance, the 503D Parachute Infantry Regiment has two battalions currently in the 173D Airborne Brigade Combat Team. The 1st Bn and 2nd Bn both wear the same Distinctive Unit Insignia on their berets, but their flashes are slightly different.
In SF units, all the members wear the same DUI, but the flashes on the Green Berets are unique down to the Group Level, which is actually higher than Battalion. More like Brigade level.