Say a user shares an article in facebook, is there any way we can include our page name and ask for a like in the same post?
Besides the obvious "Begging for likes by pestering them via personal message", you could re-share the post. Sharing a post in Facebook ensures massive coverage of the post across multiple locations, communities and often leads to a large audience responding to it, be it a positive or a negative response.
The best solution to your request is to tag your page to a comment. However, you might want to consult the owner of the original post before you do this, as he could report you as a spammer. Also, you can politely ask him to tag your page, which he can do if he edits the post.
All of this is based on the owner of the post you want your page to be tagged with. It is totally up to him. You could do this with community posts too. But since there are a large number of spammers, you could get reported or blocked if the moderators feel that you are spamming posts for personal benefits.
I wouldn't advice you to resort to the above suggestion. A bit of politeness usually goes a long way. So ask people before piggy-backing on their posts and also, in a non-irritating way, you could ask people to like your link. Cheers.