Posts cannot have Page Template features, natively, by WordPress.
They are not even native to WordPress, but provided by Themes (if you use Twenty Seventeen for example, do you see such Page Template options on pages?)
What you can do with Types, is register Post Types which can feature all of the possible things that a WordPress Post does.
This includes Page Attributes (Menu order and page parent) and Post Formats, but not Page Templates, as that is not something that WordPress adds natively, it’s something provided by a theme.
Now, I think you could profit from the rest of Toolset, as with these tools, you actually can do something similar to what you are looking for, without coding.
You may query about this here:
https://toolset.com/toolset-free-consultation/
I cannot assist (paid) Toolset features here that is why I have to redirect you in case you are interested.
However, with Types, you cannot create or choose templates on Custom Posts.
