This kind of vertical streaking is usually caused by something on the lens surface or filter, not the Milky Way itself. A smear, fingerprint, condensation, or residue on the front element can scatter bright point light sources into vertical flare.

Based on the photo update, the visible mark on the lens is the most likely culprit. First, try carefully cleaning the front element (or remove/clean any front filter). Also consider condensation: if a cold camera/lens is brought into warm, humid air and used too soon, moisture can create similar glare until it fully clears.

If cleaning the front surface and removing any filter doesn’t fix it, the issue could be on an internal lens element, in which case it’s best to test with another lens. If the problem disappears with a different lens, your original lens likely needs service.