Single player saves are usually stored locally on the console, so yes, you can keep your progress on one console and start a new save on a different one even while logged into the same PSN account.
An exception to this are games that store their save data on the developer's end, which is usually the case with online multiplayer games where you have a player profile, so your progression is associated with your PSN account and will be downloaded and synced when you start the same game on a different console.
You won't ever accidentally overwrite a save on a different console either way.
Within the same PSN account you can also copy and paste save data between your consoles by going to Settings -> Application Saved Data Management. You can either copy saves onto a USB-Stick or upload them to the PS Plus Cloud service (you need an active PS Plus subscription for the cloud option). So, if you ever do want to continue your progress for a game on a different console you also have the option to do that, but you'll have to do it manually.
If you're ever unsure or want to back up your saves, copy them onto a USB stick just in case.