$variable = STDOUT so that the STDOUT printing gets stored into the variable?

Not that way. You can however, tie a handle to a class. Fortunately for you, a module that does what you want already exists. Have a look at Tie::Handle::Scalar.

use Tie::Handle::Scalar;
{
    local *STDOUT; # Don't want to override it forever
    tie *STDOUT, 'Tie::Handle::Scalar', \my $variable;
    print "Hello, world!\n";
    print STDERR "\$variable contains: $variable";
}
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