Can you check your gadgets?

Also check your JavaScript console please. :)

Few minutes ago I've repeated again the tests: disabled local & global user scripts, disabled all gadgets (before reporting this issue I've done also this thing, just forgot to mention it).
Accessed random item, first item with one of these properties (Commons category - P373) this time was - https://www.wikidata.org/wiki/Q1325096
Again verified in more than one browser, the same issue persists - logged out is ok, logged in (under this account) the link is missing.

The console message says:

This page is using the deprecated ResourceLoader module "jquery.ui.position".  load.php:28:950
This page is using the deprecated ResourceLoader module "jquery.ui.widget".  load.php:39:794
This page is using the deprecated ResourceLoader module "jquery.ui.core".
Please use "mediawiki.ui.button" or "oojs-ui" instead.  load.php:10:432
This page is using the deprecated ResourceLoader module "jquery.tipsy".  load.php:535:171
Your skin is incompatible with VisualEditor. See <https://www.mediawiki.org/wiki/VisualEditor/Skin_requirements> for the requirements.  load.php:441:173

After some time in console apperead one more message (I hope I didn't switched to another tab and it's caused by / related to the same tab of that item page)

See also https://www.wikidata.org/wiki/Property_talk:P373#Why_not_linked.3F
Other users, @matej_suchanek and @Verdy_p also said that sometimes they encounter this issue.

I fixed T160530: [Task] Hide header image gadget when sitelinks are floated below the statement section. This one is unfortunately still valid :( Let me take a look

I found what's wrong with this. I'm finding a way the fix it atm

This would fix it. Clear the cache and have fun.

Strange. I tested and it worked for me. Anyway. Thanks for fixing it :)

Unfortunately there is no change [for me].
I think I know how you can reproduce this problem, in 'an alternative way': apend ?safemode=1 to URL and you will see (probably) unlinked values for Commons gallery and Commons category.
This newly enabled feature is a kind of safe mode for wiki user accounts.(1)
Seems that these two properties are hardly influenced by other things.

If you use safemode=1, you disable *all* gadgets including the one that makes links (Authority control. Make sure you haven't disabled it by accident) and obviously it won't work :D The proper way to test it is by being logged out which shows links to me.

Oh, God! I didn't knew that the links of these two properties are generated by this gadget. It was disabled for me for years, and with every test done to find the cause of the problem, I've disabled all gadgets...
Thank you @Ladsgroup.

It should be mentioned somewhere that disabling this gadget will leave unlinked values of P935 and P373.