tl;dr: there's a question at the end of this answer, help me answer it.
I've thought about this a lot. When sharing something on Android you can fill info in two fields:
EXTRA_SUBJECT (typically a title), and
EXTRA_TEXT (typically a message). We're currently putting the question's title in the
EXTRA_SUBJECT field and the URL to the question in the
EXTRA_TEXT field. Different apps react differently to the two pieces of data. Some just use the
EXTRA_TEXT, some use both.
The main reason we haven't put the title in the message field and just combined the two is a purely selfish one: I routinely use the "Copy to clipboard" option in the Share dialog and it not just copying the URL is one of my biggest Android pet peeves. Most other apps do what you are asking for in this scenario, copy both the title and URL to the clipboard (since that's also what would be sent over as the tweet body).
Now to make matters a lot worse, let's look at what it looks like when you try to Share to Facebook, G+, and Twitter right now:
As you can see, they all react differently to the share request. Facebook and Google+ actually ignore the fact that what we told them to share was the URL, and embedded the URL in the post instead alongside a empty by default caption for it ("Write something...) -- Twitter on the other hand does the simplest operation and flat out jams the URL we're trying to share into your tweet.
Here's where it gets worse. Let's hit "Tweet" on that draft and see what it looks like now:
Look at that! Twitter actually one-boxes/embeds the question into the tweet, while still showing the URL of it as the tweet contents, something that's not visible to you at all while you're actually composing the tweet.
Anyway, in summation: Everything is not coming up milhouse -- Sharing kind of sucks on Android. However, if we can get this question answered (post in the comments!) we'll do it:
If we change Share to share the title of the question alongside the URL to it, we'll get duplicated question titles on Facebook, Google+, and the expanded view of a tweet. We'd also have "Share to Clipboard" have both the URL and the title on it. Would this be preferred to what we have now?