Ok there are a few issues here...
You only have one input value, at very low precision. This can only contain a 1 byte value, which is a single number between 0 and 255. A 1 byte value can also be represented in binary to simulate 8 on/off switches - this is how they used it in the multiplayer tutorial. With it, they can keep track of up to 8 keys or mouse buttons that are being pressed or not.
However, consider your own needs. What does the peer need to tell the host? In your game, you'll need to send the fact that the peer clicked, and also where the peer clicked. The location is two values, an x and a y number, that can definitely go above 255. So these two values need a higher precision. I would use normal or high, since you're honestly not going to have bandwidth concerns with just two players and small amounts of data going back and forth.
So step 1, add two more input value tags, for "mouseX" and "mouseY", precision high, no interpolation.
You probably also don't need to store the peer values as instance variables in the object, since I'm assuming you may have multiple cards eventually, and either player can move them so the peer inputs are not associated with any particular card. So you can get rid of the sync variable, and use the inputs directly to keep it simpler.
-> Multiplayer: Add client input value tag "inputs", precision Very low (uint8, 1 byte), interpolation None
-> Multiplayer: Add client input value tag "mouseX", precision High (double, 8 bytes), interpolation None
-> Multiplayer: Add client input value tag "mouseY", precision High (double, 8 bytes), interpolation None
Next lets look at the peer events, to send the peer inputs to the host. Lets remove local input prediction for now, since you haven't gotten the underlying system working yet. Since you're only keeping track of 1 button press, left mouse down, you don't even have to worry about setting bits. So keep it simple. If left button is down, set inputs to 1. Else 0. You'll also always want to send your current mouse position every update, so you need to add
+ Multiplayer: On client update
-> Multiplayer: Set client input state "mouseX" to value Mouse.X
-> Multiplayer: Set client input state "mouseY" to value Mouse.Y
----+ Mouse: Left button is down
-----> Multiplayer: Set client input state "inputs" to value 1
----+ System: Else
-----> Multiplayer: Set client input state "inputs" to value 0
Your peer events should have nothing else for now. Remember the peer only tells the host what its inputs are, the host will do all the work and sync back the results to the peer.
As for your host events... the "On client update" is a peer only trigger. So basically your host is doing nothing at all right now. Again, the host needs to do the actual work of moving of the object based on peer inputs.
So first it needs to have its own way of moving the object. The drag and drop behavior should be enough for this for now.
Secondly, it needs to check if the peer's mouse is down. If it is, then is it clicking on the sprite? Then after that, if it is clicking and on top of a sprite, update the sprites position every tick to wherever the peer's mouseX and mouseY are.
At this point, if you've got everything working properly, both players should be able to move the object around with their mouse. You can try adding lag compensation after you get this working.