$str = aa;

Unless you have a variable named aa somewhere, that line will give you an error.

The rule is: $1 is used inside strings and \1 is used inside regexes.

Examine the inputs for the substitution operator:

s/regex/string/
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Here are some examples:

my $line = 'abbc';
$line =~ s/(.)\1/XX/g;
say $line;

--output:--
aXXc
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my $line = 'abbc';
$line =~ s/(.)/X$1X/g;
say $line;

--output:--
XaXXbXXbXXcX
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my $line = 'abbc';
$line =~ s/(.)/$1$1/g;
say $line;

--output:--
aabbbbcc
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I used the following regex to find repeating characters in a string.

$str = aa;
$str =~ /(a)$1/;
print "$&";
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This does not match 'aa'. It matched only 'a'.

A regex acts like a double quoted string with respect to variable interpolation (and with respect to string escape sequences). $1 has not been assigned a value by a previous match, so $1 is equal to undef. And when undef is used like a string, perl converts it to a blank string. Interpolating undef into a double quoteish context, like your regex, means you are interpolating a blank string, so your regex is equivalent to: /(a)/