“This is the first transformer I’ve seen like this.”
I’ve seen transformers before, but none quite like this. This was mounted on the opposite side of the home as the front door, door bell, and chime unit. In this case, I’m absolutely sure this one was not connected to those units.
This home did however have an old RJ 4 Prong telephone jack. Brian mentioned they may have been used for old phones, so maybe it was related to that? Or maybe even something that predated that? Regardless it was already written up based on other factors.
The green transformer in the photo is slightly different because it does not have three conductors as shown in your graphic. The third conductor is an EGC to ground the transformer (more modern transformers have the EGC so that they can be connected to plastic boxes). The green transformer is grounded by connection of the threaded portion with the locknut. That would be the 120 volt side. The green transformer is required to be grounded and it is not grounded since it is not attached to a grounded metal box or metal panel enclosure, etc. Also the 120 volt connections cannot be in free air.