Same WIRELESS mouse + keyboard (Logitech Unifying) for two different computers?

I'm going to answer no - it is not possible. There appear to be similar questions on Logitech's site so you can search there as well. I suspect the answer is that when you pair, the transceiver and the device exchange several bytes of random data and form a key to both allows the devices to recognize each other and to encrypt their wireless transmission.

The problem is that the transceiver may have the ability to recognize something like 5 devices, but the input devices can only recognize and pair to one transceiver. That's because the transceiver only has to sit there and accept a signal it recognizes, otherwise it is passive. Sit, if you see a signal with the correct encryption and ID, accept input, otherwise ignore.

If you added the ability to have the input devices (mouse, keyboard...) pair to multiple transceivers, you would require some sort of additional buttons on the device to switch between them. You'd also need to have the input device store the state (caps lock, num lock...) it was in for each transceiver.

The only Logitech device that can do this is the new K760 keyboard for bluetooth. None of the Unify devices appear able to do this.

The new Logitech M720 has this feature:
http://www.logitech.com/en-us/product/m720-triathlon

How I use it:

• Channel 1: Notebook (Unifying receiver)
• Channel 2: Android Smartphone (Bluetooth)
• Channel 3: Desktop PC (Unifying receiver - not included in box content)

Alternatives:

1. Some monitors have a built-in usb switch. Search for a monitor with two usb upstream ports: https://www.google.com/search?q=monitor+"two+upstream"
2. You could buy a KVM switch to switch multiple ports like video output, sound and USB or you could buy a usb switch (the cheapiest method).
3. Use for Windows the Mouse without Borders tool to use one mouse with multiple windows PCs:
   https://www.microsoft.com/en-us/download/details.aspx?id=35460
   Or use Teleport if you prefer Apple:
   http://abyssoft.com/software/teleport/
4. Buy a M720 and combine it with 3.) so you can control all your Apple and Windows devices with one mouse and keyboard ;)

I solved this, to some extent, with AutoHotkey and figuring out what causes the de-pairing. This solution requires some seconds for re-pairing each time, but is fast enough to be a useful method.

The AutoHotkey code:

F2::
  Run, C:\Program Files\Common Files\Logishrd\Unifying\DJCUHost.exe ; this is the Unifying connection program
  WinWaitActive ahk_exe DJCUHost.exe,, 4
  If !ErrorLevel
  {                                            
    Sleep 160 ; 100 was not enough
    Send {Enter}
    TrayTip Re-pairing Logitech,To disconnect: pull dongle, hibernate or use a USB switch to disconnect the dongle
    ; As soon as you have re-paired, you can switch on (or re-insert or wake the computer with) the Unifying dongle to put it back into operation (it then waits for a missing pairing)
  }
  else
    TrayTip Could not open the Unifying connection program,`n
  Return

Basically that code is only opening the Unifying connection program, but speeds things up. The thing is to understand that the pairing is (temporarily) lost as soon as power is lost for the small Logitech Unifying dongle. Then, a new pairing can be done quickly. You can achive the required power loss by: 1) pulling the dongle out, 2) using a USB hub with simple toggle switches (or a KVM Switch), 3) putting the computer into hibernation, after which the dongle loses power (you can't have the dongle on a permanently powered USB hub for this to work, also just sleep might be enough if you set your computer to cut the power to the dongle during sleep). (There's also option 4, you can go to Logitech Unifying Software: Advanced, Un-Pair. But that can't be done blindly, though in some cases a script could do that un-pairing.)

So, using the option 2 or 3 is just more convenient than removing the dongle from the computer that needs to get un-paired, especially if your setup is like mine where the dongle is located in a hard to reach place.