So I tried creating a magic item that should set the Current DEX to 23 and the current Con to 5. This was made as a wondrous item so that the character can equip it.

The result was that my character ended up having 23 in DEX - but con did not change at all. I deleted the item and retried it because I believed I had made a mistake along the way somewhere. New item, same rules, same result (Fortunately). Having tried with different amount of  "Set" to see what happens, and - it simply only takes the first modifier and ignores everything else, is this "Working as intended"? Have I forgotten to check the secret checkmark, or is this a bug? ;-). 

It could be a server caching issue.  Here's an approach to try:

Make sure you have saved your homebrew item using the topmost save button (in the Basic Information section)

Lot out of your account; log back in; wait 20 minutes or so.

Then try adding the item and see if it works correctly.

Have done that :) been offline for over 10 hours now. Added the item, same result.

The item itself states quite clearly

Notes: Set: Dexterity Score, Set: Constitution Score

From the Item itself:

Set - Dexterity Score 23

Set - Constitution Score 5

If I change the item and have the Con score first, the item takes effect immediately, but still only the first modifier is applied regardless.

Replicating with a new item:

• Create Magic item from scratch
• Give it a name
• Set rarity to something (I chose Very rare)
• Base item type: Item
• Magic item type: Wondrous item
• Add something for description
• Press create item
• Press + for Add modifier
• Modifier type: Set
• Modifier subtype: Dexterity Score
• Fixed value: 23
• Press Save
• Press + for Add modifier
• Modifier type: Set
• Modifier subtype: Constitution Score
• Fixed value: 5
• Press Save
• Press Save on the item

[OPTIONAL]

• Wait 20 minutes or so..
• Add it to character
• Equip Item
• Note that Dexterity goes up, but Constitution stays where it is.

This happens every time I create a new item with 2 or more Set in it. Only the first is ever used.