Somehow I missed your response until now, game-ryo! I see what you did. That’s useful to understand those settings. After reviewing your comment and at least getting my sites to show up in the dropdown (without ajax), I went back to focusing on how to get the ajax set up properly. I returned to another implementation I’d had trouble with previously, from JamesImpression: https://github.com/JamesImpression/acf-multisite-select
I was able to combine that code’s javascript with my php code based on post_object and finally got it to work!
The key was to register and enqueue the js (in assets/js/input.js) that calls acf.registerFieldType. It also allows for additional initialization code, but I didn’t need that.
I’ll try to keep an eye out for alerts if you see this, want to give it a try, and have any questions.
I am trying to do something similar and am having the same issue. I suspect that the issue relates to the javascript and jQuery that should be replacing either box with a search & select (jQuery select2) field. I noticed that a post_object field’s select box gets the class “select2-hidden-accessible” and an extra span with classes “select2” and “select2-container”. My custom object multi-select does not get that class nor the additional span element.
As for the “number box”, that is just what a multiple select field looks like when it is a single line. The arrows allow one to scroll through the options to select/deselect them.
