App Store no longer updates apps
I have been having the same issue on Mojave 10.14.6 with trying to update apps from the App Store, with installs of new apps (including if you delete an app you're having trouble updating, then just install it again) working fine.
For reference of others reading this, most of my Apps (certainly the ones affected) are just in /Applications so that's not part of the cause.
Probable Cause
Searching around, I found that the community has identified an issue:
https://mrmacintosh.com/mojave-2020-005-security-update-causing-major-problems-updated/
The TLDR version is that:

if you installed Safari 14
then the most recent Security Update

in that order (Safari 14 first), then you tend to encounter the issue; if you installed the Security Update first, then Safari 14, you're likely to be ok.
Checking if you're affected
The last part of the TLDR is that you can check if you're encountering the same issue by running the following in the Terminal:
softwareupdate -l
If you are, you'll see the following output:
objc[5711]: Class AMSupportURLConnectionDelegate is implemented in both /System/Library/PrivateFrameworks/EmbeddedOSInstall.framework/Versions/A/EmbeddedOSInstall (0x108f37c58) and /System/Library/PrivateFrameworks/OSPersonalization.framework/Versions/A/OSPersonalization (0x108d9a358). One of the two will be used. Which one is undefined.

So what do I do now?
It seems Apple has pulled the updates, so we can hope there will be some fix and revised update for everyone who hasn't installed the updates yet soon .
The current solutions (detailed on the above link), other than waiting, are that you can do a full re-install of macOS 10.14.6 - over the top, and then go from there.
Installing Catalina is another option, though that may not solve all issues, especially as the latest Catalina patch is also causing issues for some: https://mrmacintosh.com/catalina-10-15-7-update-accountsd-using-400-if-using-icloud-mail/
I'd read and keep an eye on the above link as it gets updated with latest best practice ways to resolve this, and when Apple release an update that resolves things - though it's probably best to wait and see the reaction to that, when it does appear.
Hope this helps!
UPDATE: (1st Oct 2020 13:39 UTC)
Just tried again offhand and now all the updates installed. I recall some server-backend issue like this before a few months ago.
Still doesn't get around the other issues mentioned in the above link, but at least app-updates now install.
Also, thanks to @bmike for the link below in the comments and his separate reply - anyone in future coming here, please check those out.
UPDATE: (2nd Oct 2020 10:11 UTC)
...And the problem is back. Today updates again won't install.
I guess just going to play the waiting game for a bit longer and see what happens.