<edit>
If you're being prompted to close an application that uses Flash, then you're attempting to install the version that already exists on the system. Flash first attempts to uninstall whatever version exists on the system before installing, if the version on the same is the same version being installed, and it's in use (e.g. loaded by the browser), then the existing version can't be deleted and the user is prompted to close the application for installation to proceed. If Flash doesn't exist on the system or in an upgrade scenario, Flash can be installed without closing the browser
<end edit>
It's possible some other application is identifying itself as Chrome.
With that said, Google embeds Flash Player in Chrome. There is no need to download it separately, unless there are issues with the embedded version. Google does block Flash by default, forcing users to enable it. Due to this, oftentimes, content will not detect that Flash is installed and prompt users to download and install it.
To enable Flash in Chrome, see Use or fix Flash audio & video - Computer - Google Chrome Help
Note that as of Chrome 69, Chrome deletes the user's 'Allow Flash' settings that they've added when Chrome is closed, forcing users to enable it when they launch Chrome again. This is a Chrome implementation, not an issue with Flash Player. Per Google's Flash Roadmap - The Chromium Projects :
Non-Persisted HTML5 by Default (Target: Chrome 69 - September 2018)
Summary
Sites using Flash will require explicit permission to run, every time the user restarts the browser.
Rationale
Require affirmative user choice to run Flash Player content, without that choice persisting across multiple sessions.
2
Replies
AdChoices