Looks like the BMW E36 (325i 1995yr) hub broke at the axle spline.
https://www.napaonline.com/en/p/NDP7707822 (L-R hubs are the same PN)
Was there a nut on the hub (inside the pin drive hub)?
I'd change the axle 1/2 as well ( L-R axles are the same PN)
https://www.napaonline.com/en/p/NMD951209
https://www.napaonline.com/en/p/ATM33411132565
Yes they can have Open and LSD diffs.
The aft cover of the diff should have a metal Tag. If the Tag starts with an "S" its an LSD (followed by the ratio)