educate me on my vintage Longines watch

I see no sign of it being military issued. What makes you think that?

Sellers often just say that watches are military. Military issued watches from that era almost always have some distinctive markings on the back, specific to the military to which they are issued.

As @Dan S mentioned, there are no indications of this being a military watch. Further, the dial does not appear to be original. The hour markers look strange, and inconsistent with original dials. The font in the outer track is also not consistent with original dials (e.g., the shape of the 5s). Also, the white printing is atypical, as black dials from this period were mostly made using a process that yielded either silver or gold printing. From the photos, it almost looks like the dial is a metallic blue, which is certainly not correct for such a watch. Below are two examples from the same order (22'671) with original dials. Your watch likely started its life with the same type of dial. Additionally, your watch has an incorrect crown, incorrect ratchet wheel screw, and what appears to be a re-chromed mid-case. So all in all, a watch that has been heavily modified.

Source: https://marketplace.watchcharts.com...s-vintage-ww2-military-era-watch-cal-10l-1943


Source: https://wannabuyawatch.com/product/longines-manual-wind-usn-buships-circa-1940s/