In the Windows Search slot, type "Controlled" and select the result "Controlled folder access" and click it.
Turn off Controlled folder access.
See if the screen shot function returns to normal.
If it does, great, leave the Controlled folder access switch off.
If you would rather still use the anti-ransomware feature, then have the switch on and use the "Allow an app through Controlled folder access" link, click the + box for "Add an allowed app", then navigate to the FF14 game client install location and add the game .EXE file.
The anti-ransomware feature was added to Win10 in the past year or two, but Microsoft is slow to add "default whitelisted" programs to it's built in definitions, so it's hit-or-miss as to if the exact FF14 you have is pre-OK'd. This is why not everyone has the issue with screen shot saving function, as they either have an updated whitelist, custom set their FF14 screen shot folder outside of Window's protected folders, don't run the Windows protection, or have a game client version that was OK'd.
If you can't adjust the settings on the protection, then your windows user account truly isn't at Administrator permission levels, which will also result in the screenshot function of the game to not work.