I also stumbled across this.
After selecting the default skin, in the settings-page (https://ctan.org/user/settings) the various skins were displayed/shown/listed, but none of them could be clicked to be applied.
Seems the problem occurs only while the default skin is in use.
(Maybe the problem is related to script- and/or add-blockers in use.)
Here's how I got around it:
Method 1:
In the Internet browser, delete all cookies belonging to https://ctan.org.
(Chromium browser: Browse to the settings-page, https://ctan.org/user/settings, and click with the right mouse button on the window of the Chromium browser where the page is displayed. A context menu pops up. In this context menu, at the very bottom, with the left mouse button click on "Inspect". (You can also use the shortcut ctrl-shift-i.) The window splits in two. On the left side the web-page is still displayed as usual. On the right side you see the source code of the page and tables with lists of properties of elements to be examined. The header of the right side consists of a horizontal menu bar—items of that menu bar are: Elements/Console/Sources/Network/Performance/Memory/Application/Security/Lighthouse. There is a menu item "Application". Select it by clicking with the left mouse button. (Maybe you have to click the » sign in the menu bar with the left mouse-button to get this menu item displayed.) The menu "Application" opens up. Under "Storage" click the item "Cookies" with the left mouse-button. Thereupon a list with entries folds out downward. In this list, right-click the entry "https://ctan.org". In the context menu that opens, with the left mouse-button click on the entry "clear". Then close the right side of the split window by clicking with the left mouse-button on the X symbol at the right of the menu-bar where "Application" was clicked.)
Reload the settings-page https://ctan.org/user/settings.
Usually you will see the Gray theme again.
On Halloween you will see the Halloween theme again.
With these themes all skins on the settings page are clickable and selectable.
After scrolling so that you see the top of the page, at the top right you find a menu-bar Login/Join/Settings/Help.
In this menu-bar click on "Settings" with the left mouse button.
Now you can choose a skin. E.g. under the heading "Color-based Skins" the skin "Gray".
Method 2:
In the cookie of https://ctan.org change value of the key skin, which is empty when the default-skin is in use, to gray.
Chromium browser: Browse to https://ctan.org, and click with the right mouse button on the window of the Chromium browser where the page is displayed. A context menu pops up. In this context menu, at the very bottom, with the left mouse button click on "Inspect". (You can also use the shortcut ctrl-shift-i.) The window splits in two. On the left side the web-page is still displayed as usual. On the right side you see the source code of the page and tables with lists of properties of elements to be examined. The header of the right side consists of a horizontal menu bar—items of that menu bar are: Elements/Console/Sources/Network/Performance/Memory/Application/Security/Lighthouse. There is a menu item "Application". Select it by clicking with the left mouse button. (Maybe you have to click the » sign in the menu bar with the left mouse-button to get this menu item displayed.) The menu "Application" opens up. Under "Storage" click the item "Cookies" with the left mouse-button. Thereupon a list with entries folds out downward. In this list, left-click the entry "https://ctan.org". In the table with columns "Name", "Value", etc select the row where the column "Name" holds the value "skin" and in that row in the column "Value" type "gray" (without the surrounding quotes).
Then close the right side of the split window by clicking with the left mouse-button on the X symbol at the right of the menu-bar where "Application" was clicked.
Reload the page https://ctan.org.