Actually you don't even need a regex for this:

$s{substr($_, 2,index($_,'/',2)-2)}++ for <DATA>;
print reverse sort map {"$_ = $s{$_}\n"} keys %s;
__DATA__
//word.one.team/other.stuff/Info.info
//this.sentence/other.inforation/moreInfo
//first.part.here/set.list
//this.sentence/data/processing
[download]