I appreciate Toby and William commenting that they are still seeing this warming.  All of my physical Ubuntu boxes are as well, despite having all of the specified upgrades performed.  I thought perhaps I was losing my mind, but not quite yet.

I even removed all of the mysql and mariadb related packages, reset, and re-installed just the current revisions -- and still the warning persists.

However, I spun up a fresh Ubuntu 22 virtual machine and installed the mysql client packages, and that virtual system produces no warnings.  All of the package versions are identical to those on the machines on which the warnings are still produced -- which doesn't make any sense, but there you go.

Machine A:
user@machineA:~/Code/db$ ./connect.pl 
WARNING: MYSQL_OPT_RECONNECT is deprecated and will be removed in a future version.
$VAR1 = bless( {}, 'DBI::db' );
user@machineA:~/Code/db$ ./versions.sh 
ii  libdbd-mysql-perl:amd64                       4.050-5ubuntu0.22.04.1                      amd64        Perl5 database interface to the MariaDB/MySQL database
ii  libmysqlclient21:amd64                        8.0.34-0ubuntu0.22.04.1                     amd64        MySQL database client library
ii  mysql-common                                  5.8+1.0.8                                   all          MySQL database common files, e.g. /etc/mysql/my.cnf
ii  libdbd-mysql-perl:amd64                       4.050-5ubuntu0.22.04.1                      amd64        Perl5 database interface to the MariaDB/MySQL database
ii  libmariadb-dev                                1:10.6.12-0ubuntu0.22.04.1                  amd64        MariaDB database development files
ii  libmariadb3:amd64                             1:10.6.12-0ubuntu0.22.04.1                  amd64        MariaDB database client library
ii  mariadb-client-10.6                           1:10.6.12-0ubuntu0.22.04.1                  amd64        MariaDB database client binaries
ii  mariadb-client-core-10.6                      1:10.6.12-0ubuntu0.22.04.1                  amd64        MariaDB database core client binaries
ii  mariadb-common                                1:10.6.12-0ubuntu0.22.04.1                  all          MariaDB common configuration files



Machine B:
user@machineB:~/Code/db$ ./connect.pl 
$VAR1 = bless( {}, 'DBI::db' );
user@machineB:~/Code/db$ ./versions.sh 
ii  libdbd-mysql-perl:amd64                       4.050-5ubuntu0.22.04.1                      amd64        Perl5 database interface to the MariaDB/MySQL database
ii  libmysqlclient21:amd64                        8.0.34-0ubuntu0.22.04.1                     amd64        MySQL database client library
ii  mysql-common                                  5.8+1.0.8                                   all          MySQL database common files, e.g. /etc/mysql/my.cnf
ii  libdbd-mysql-perl:amd64                       4.050-5ubuntu0.22.04.1                      amd64        Perl5 database interface to the MariaDB/MySQL database
ii  libmariadb-dev                                1:10.6.12-0ubuntu0.22.04.1                  amd64        MariaDB database development files
ii  libmariadb3:amd64                             1:10.6.12-0ubuntu0.22.04.1                  amd64        MariaDB database client library
ii  mariadb-client-10.6                           1:10.6.12-0ubuntu0.22.04.1                  amd64        MariaDB database client binaries
ii  mariadb-client-core-10.6                      1:10.6.12-0ubuntu0.22.04.1                  amd64        MariaDB database core client binaries
ii  mariadb-common                                1:10.6.12-0ubuntu0.22.04.1                  all          MariaDB common configuration files


There's something else effecting this database connection, but I don't know what it is either.

edit: For those using DBD::MariaDB, make certain that you are running v1.23+.  1.22 causes this warning to be displayed, even if all of the Ubuntu packages are up-to-date, but 1.23 apparently contains a patch which squelches these warnings.  I just updated a few of my physical Ubuntu machines from 1.22 to 1.23, and disappeared these annoying warnings.