You're confusing string literals (code) and strings (values).
In the source, "$2$1" is a string literal. It results in a string made up of the values of $1 and $2.
my $p = "$2$1"; # my $p = $2.$1;
# Doesn't change the value of $2 or $1.
In the source, "$ARGV" is a string literal. It results in a string made up of the value of $ARGV.
my $p = "$ARGV"; # my $p = "".$ARGV;
# Doesn't change the value of $ARGV.
If you want process the value of a variable for $1 and $2 and replace them, you are trying to implement a templating system. There's a one that matches your specs called String::Interpolate.
use String::Interpolate qw( interpolate);
my $t = '$1$2';
my $t = $ARGV;
my $p = interpolate($t);
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