An easy way is to split on '/' which ordinarily needs to be escaped. The resulting array has nothing in the first two slots,

my %count;
$count{(split /\//)[2]}++ while <DATA>;

printf "%s = %d\n", $_, $count{$_} for keys %count;

__DATA__ 
//word.one.team/other.stuff/Info.info
//this.sentence/other.inforation/moreInfo
//first.part.here/set.list
//this.sentence/data/processing
[download]

Update: ++tos for the correction. Repaired.

After Compline,
Zaxo