I am using CPT-onomies with ACF (free version). I believe to make them work together correctly, you have to both assign the CPT-onomy (as you would a category or a tag) to the post AND create a taxonomy ACF that uses values from the CPT-onomy and is assigned to the applicable post types. Then use the field to choose one or more values from the CPT-onomy while on the edit-post screen.
Then you have to create custom templates and/or write functions to get your custom fields to appear in your posts. Both Elliot and Rachel provide ample documentation and examples on their websites so you can find the proper functions to use in order to accomplish that.
By the way, you have to create your CPT-onomy posts BEFORE you can assign them. The Add New functionality doesn’t work from the edit-post screen like it does for categories and tags.