You might get a better (more definitive) answer from the Joomla community than the TKL community, but I'll give it a go.

I haven't played with Joomla for a while, but I think you can:

1. duplicate the joomla directory for the second site (if the default location is /var/www/joomla, you can copy it ands the contents to /var/www/secondsite)
2. Modify the second sites configration to use a second database to store content
3. Modify the apache configuration to create a second virtual host as you note above

There may be a few other gotchas but I think you have the right idea and have started down this path. Have you tried this to see what happens?

I'm not sure, but...

I'll try this out a little later today and post back my results.

In order to test this, I just created a second site in my apache configuration and pointed it at a new directory containing just in index.html file.

Presumably, I could copy in the joomla files, create a second database in mysql, and change the configuration.php settings and have things working. I didn't got that far in this test

Here is what I did

• As root, logged in and created an additional file in /etc/apache2/sites-available called "site2" to represent my second site. I actually just coped the "joomla" site file to "site2" and edited it as appropriate and added the "ServerName" directive.
• Create a symlink in /etc/apache2/sites-enabled to the /etc/apache2/sites-available/site2 configuration
• Restarted apache2

After doing that, http://site2 brought me to my new site and anything else showed joomla.

For reference, here is my /etc/apache2/sites-available/site file

NameVirtualHost *:80
NameVirtualHost *:443

<VirtualHost *:80>
    ServerName  site2
    UseCanonicalName Off
    ServerAdmin  webmaster@localhost
    DocumentRoot /var/www/site2/
</VirtualHost>

<VirtualHost *:443>
    SSLEngine on
    ServerName site2
    SSLCertificateFile /etc/ssl/certs/cert.pem
    ServerAdmin webmaster@localhost
    DocumentRoot /var/www/site2/
</VirtualHost>

<Directory /var/www/site2/>
    Options +FollowSymLinks
    AllowOverride All
    order allow,deny
    allow from all
</Directory>

That should do what you need but please post back as to your success/failure.

Here are my two files:

/etc/apache2/sites-available/joomla (I did not modify this at all)

NameVirtualHost *:80
NameVirtualHost *:443

<VirtualHost *:80>
    UseCanonicalName Off
    ServerAdmin  webmaster@localhost
    DocumentRoot /var/www/joomla/
</VirtualHost>

<VirtualHost *:443>
    SSLEngine on
    SSLCertificateFile /etc/ssl/certs/cert.pem
    ServerAdmin webmaster@localhost
    DocumentRoot /var/www/joomla/
</VirtualHost>

<Directory /var/www/joomla/>
    Options +FollowSymLinks
    AllowOverride All
    order allow,deny
    allow from all
</Directory>

/etc/apache2/sites-available/site2

NameVirtualHost *:80
NameVirtualHost *:443

<VirtualHost *:80>
    ServerName  site2
    UseCanonicalName Off
    ServerAdmin  webmaster@localhost
    DocumentRoot /var/www/site2/
</VirtualHost>

<VirtualHost *:443>
    ServerName  site2
    SSLEngine on
    SSLCertificateFile /etc/ssl/certs/cert.pem
    ServerAdmin webmaster@localhost
    DocumentRoot /var/www/site2/
</VirtualHost>

<Directory /var/www/site2/>
    Options +FollowSymLinks
    AllowOverride All
    order allow,deny
    allow from all
</Directory>

Are you accessing your site with the site2 url? You need to write your client's host file to point to the appliance id and access with the ServerName url in order to get to the second site. 

You could also check if there's a .htaccess file rewriting url's to the first site, and last, check the DB of joomla which probably will need to update a value with the site's url (at least wordpress needs this). 

Just some ideas reading quickly your post.