I think the Marines have this to a limited extent within some units (notably the components of the First Marine Division, though there is always that overarching "Marine" sense that can eclipse individual unit ties. The Navy tries (with some success) to duplicate this with ship ties and the continuation of squadron and carrier wing histories and traditions.
You're exactly right. I've spent my operational tours as an officer within the 1 MARDIV, and my preference would be to return there for the simple fact that it is what it is. For those who fought under Gen Mattis when he as the commanding general, many of them would probably tell you that if he said, "the enemy is at the bottom of this cliff, and I want you to jump off of it and kill them," they would do so unhesitatingly. It's an exaggeration, but that's the sense of bond he built within the unit.

In a Regimental system, reputation stands at the forefront, because you "came up through the ranks", and even if you leave for a stint elsewhere, upon your return there will be folks who remember you. I think that is empowering from a leadership standpoint, because subordinates have a better grasp of your background, capabilities, and past performance. If you were known to be good for your word and led well, subordinates will have more confidence in your orders to them, even if theyve yet to be tested in combat. I'd say this is dramatically so in formations like the Ranger Regiment and SFOD-D.

As for sigils, they are indeed powerful. The 3d Light Armored Recon Bn goes by the tactical callsign of "Wolfpack", and in OIF I, every vehicle bore a wolf's head on its turret, as well as a a symbol for its respective company. Folks knew who we were just from a quick glance, and our Marines tended to walk with a perceptible swagger.