Prior to today, I used the following URL to get the Facebook share/like/comment count of a URL:
https://api.facebook.com/method/links.getStats?format=json&am
Looks like a bug in fb-api.
My solution:
receive the number of shares/comments as you describe in your question with
graph.facebook.com/?fields=og_object{id},share&id=https://stackoverflow.com/
save shares/comments count
save fb object-id of url
og_object->id
get likes count with (max limit is 1000, then you can use paging):
graph.facebook.com/OBJECT_ID/likes?access_token=ACCESS_TOKEN&pretty=1&limit=1000
UPD 2016-08-22
I found a solution that allows you to get the number of likes/reposts/comments in one step:
https://graph.facebook.com/?fields=og_object{likes.limit(0).summary(true)},share&ids=http://google.com,http://twitter.com
It is not currently possible. Facebook is adding likes/shares/comments for URLs and returning them as shared_count. Not possible to return correct number of individual likes/shares/comments, and that is not clear in the Graph API documentation. Not clear if it is a bug or a documentation error.
The old Rest API that provided this data was turned off on the 18th August.
There is a comment from a Facebook engineer explaining this in reply to a bug report, in the answer to this Stack Overflow question: Getting Facebook Share, Like and Comment Counts for a Given URL with API Graph v2.6
You can also subscribe to this bug report at Facebook, but is a bit old with still no solution: https://developers.facebook.com/bugs/748651458568287/