Games installed via the Itch.io app typically store their saves in a directory within the app's data folder.
English is not very precise here. Which app do you talk about?
Because, well, the itch app and the installed games do not care either way. Itch is not Steam. The app only downloads the stuff for you, there is no "install" in most cases. It is unzip and run. It does not matter if the app does this, or if you do this with your browser.
Especially with renpy games. All renpy games I know, are "portable". They do not care, where the executeable is. They will save their save game in two locations. That user folder in appdata and the game folder as a backup.