If a user clicks on:
Users > their name > show user posts
they get: "This user has made no posts."
or
My Stuff > My Posts
they get: "This user has made no posts."

But if the admin or another user clicks on:
User > name > show users posts
all of the posts show up.

How can a user see all of their own posts?

We have UBB.threads™ 7.5.4.1

Humm?
Is this for more than one user?
You might need to have the user rebuild their cookie,
Or use the rebuild content in the control panel.

There are several options for rebuilding content. Which option should be chosen?

Well you could just start from all of them.
But my statement is if you can see the posts but the user can't then I would suspect that it is a problem on the user part. meaning maybe they need to rebuild their cookies.

Try becoming the user and see if you can replicate the issue.

I became the user and did duplicate the issue.

The user has deleted his cookies yet the problem persists.

Any advice?

Well, if you duplicate the issue it's not the users cookies, it's something with the forum; to which I bring you to the previous suggestion of rebuilding items...

Thank you Gizmo and Ruben. I will let you know how things go.

I rebuilt everything that can be rebuilt.

The user himself has deleted his cookies.

If a user (or I logged in as the user) clicks on:
Users > their name > show user posts
they get: "This user has made no posts."
or
My Stuff > My Posts
they get: "This user has made no posts."

But if the admin or another user clicks on:
User > name > show users posts
all of the posts show up.

I am at a loss as to what to do next to fix this.

I assume that you used:
Rebuild Post Counts

So since I am at a loss for a suggestion.
I would next manually delete the session files as well as the cache files for extra measure.
Meaning don't delete the htm or html files in either folder.
Then rebuild cache and see if that works.