This what the docs say:

if the module Apache::DBI has been loaded. In this case every connect request will be forwarded to the Apache::DBI module. This looks if a database handle from a previous connect request is already stored and if this handle is still valid using the ping method. If these two conditions are fulfilled it just returns the database handle. The parameters defining the connection have to be exactly the same, including the connect attributes! If there is no appropriate database handle or if the ping method fails, a new connection is established and the handle is stored for later re-use.

and:

it keeps database connections persistent on a per process basis. The problem is, if a user accesses several times a database, the http requests will be handled very likely by different servers..

But if your requests are dealt with by the same process (which is very likely, as it is most difficult to open many simultaneous requests by hand (hint: use a stress test tool)) you use the same connection.