Yes, technically this is possible. Your main concern will be publishing exactly the same content to both of these sites.
Normally with two separate sites (from the same Tridion Publication), you would have two separate Tridion Deployers and two separate Broker databases. However, as you say, you want to ensure that the content is exactly the same on both of these sites, and have a single, shared Broker database. Therefore, to avoid database update conflicts (with two Deployers trying to write the same thing to a single database), you should only have a single Tridion Deployer.
Unfortunately, there is no way for a single Deployer to write the same files to two different locations on the filesystem (one location for each of your websites).
To get around this, I'd propose that you have a single website set up as normal (with a single Deployer writing to the Broker database) and then use Robocopy or XCopy (or similar) to monitor the folders of Site 1 (excluding your
web.config) and then automatically copy any changes to Site 1 (when an item is published or unpublished) to the same location in Site 2.
Alternatively, you could publish to a shared filesystem location and have both of your sites use this same single location. However, you would need to implement a mechanism for having separate configuration files for each of you sites separately with this approach. You may also get issue with locked resources that are shared too.
Other things that you may need to consider include:
Does Experience Manager need to be configured for both of these sites?
You will need to ensure that you application code is exactly the same on each of the sites (other than the config that you mention).
To help with debugging, you should configure the logging (both Tridion and the application) to write to different locations