Do you have a preference to for design or development (or neither, or other…)?
Web Design: Site design, user interface, interaction, motion graphics, UX, etc.
The logic and programming skills from game dev should serve you well as a developer. Smaller companies usually offer the freedom to do both, but you’ll need to specialize somewhat at most enterprise-level businesses. I work with a few Game Dev majors; they tend to be good developers, because game dev has (at least historically) a lot to do with solving for constraints. Web dev is just a different constraint set.
I’ve worked at big and all companies. It’s nice to have freedom of small companies to manage lots of different aspects. It’s also nice to have the freedom of large companies to focus on an area of expertise.
Weird big dreams are good. If you want to work for a fashion company, figure out what the roles look like and work your way back from there to what you need to do next.