I am running Joomla 3.4.1, no special admin template, no view overrides, no style overrides, but my multi-select looks like a MULTI SELECT dropdowb box as opposed to your fancy item picker

Here are your vs mine screenshots

Yours



Mine

You are missing the chosen library.
It is strange because as far as I know it is shipped with Joomla. Isn't it ?

It works the same, but the rendering is different. It is not Cook problem.
The only diffirence comes with the list groups wich are handled differently with chosen. (Causing problem with second level of group)
But that's not the point.

I just installed in another Joomla install 3.4.3.. and the fields render perfectly.. wierd!

OK, local issue, sorry!