It's not available as an .exe file. It opened in my browser.
Just so I understand, why were you looking for an EXE/stand-alone version of the game?
You're correct that there is no EXE; SIC-1 is a browser-based game for now. A desktop version could probably be built using Electron, but I don't have plans to do so. Launching from the Itch app sort of does this (it should open a new window that, while a web view, isn't just a normal browser window); it might even work offline (although all the leaderboards and whatnot won't work since they require Internet access).
Will my progress on the app be saved and will I be able to see my progress in my browser?
Sadly, no. The workaround is to just copy-paste your solutions.
I’m hoping to implement that eventually, but I haven’t done the work yet and there’s a bug in the Itch app that prevents it from working seamlessly anyway (although I have an active pull request with them on GitHub to fix that part).