Q: Magic Mouse 2 only connects with basic functions

Is the other Macbook on El Capitan, the Magic Mouse 2 is only compatible with El Capitan and beyond.

Eau Rouge  the included : "MacBook Pro (15-inch Early 2011), OS X El Capitan (10.11.5)".

For original poster:

- Try resetting the SMC and NVRAM/PRAM

Intel-based Macs: Resetting the System Management Controller (SMC)

https://support.apple.com/en-us/HT201295

About NVRAM and PRAM 

https://support.apple.com/en-us/HT204063

- Try starting in Safe Mode

OS X: What is Safe Boot, Safe Mode?

https://support.apple.com/en-us/HT201262

- Boot to Recovery and repair the startup disk

OS X: About OS X Recovery - Apple Support

https://support.apple.com/en-us/HT201314

- Try deleting the mouse .plist file

Deleting the System Preference or other .plist file

- If it is repairable reinstall the OSX

How to reinstall OS X on your Mac - Apple Support

https://support.apple.com/en-us/HT204904

- If you do not have a backup use disk utility to restore the internal disk to an external disk so you can try to recover data.

Then format the startup disk and do a fresh install of OSX

Just what did Apple on-line say?

The Apple page just says for the Magic Mouse 2 El Capitan and Bluetooth.

For the Magic trackpad 2 you also need El Capitan but also Bluetooth 4.0 which your Mac is too old to have

Magic mouse 2 will ONLY function properly with El Capitan.  It says that very clearly on the box.   Can I repeat the first sentence.   "Requires bluetooth enabled Mac with OSX 10.11 or later.'   There are no ifs or buts.