Interesting. So a few things to note - Netlify Dev is connecting to something on port 8080. I’m pretty sure it won’t spit out the big “Server now ready” box unless it does.
Oddly enough though, I don’t think it’s your Vue development server it’s connecting to. I wonder if something else (maybe another service on your machine, maybe a different instance of your Vue dev server that got orphaned or lost somewhere?) is using port 8080. I ask because if you look at the bottom of the log where Webpack finishes building then Vue says “App running at:” it’s actually running on port 8081. 
I’m not a Vue expert but I think that only happens when port 8080 is already taken.
Short of digging down the rabbit hole of what’s using your ports, can you try running
netlify dev --targetPort 8081
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Jon