You are assuming OBS references the device the way other programs do.... not a good assumption. And Logitech is NOT known for quality drivers (/heavy sarcasm). Logitech could have some crap left in the Registry causing problems for OBS [or not]. my point... your having problem with only OBS, doesn't mean its OBS' fault [it might be, it might not be. Let go of any frustration towards OBS, and your likely to get working sooner, regardless of fault]
did you try what I said, remove and re-add audio source?
Also, are you using Logitech microphone from default/system audio settings, or as a scene/source specific Audio Input? I was having audio issues until I followed the recommendation to DISABLE all default desktop audio, and then specifically enable only that desired within scenes (which works great for me)
So... if you are trying to get your system working, I repeat my recommendation from above, as well as if that doesn't work, removing Logitech microphone audio source from OBS, then deleting/rebooting/re-installing logitech driver, reboot again, then try adding audio source back into OBS.