> What am I doing wrong?

you have multiple issues

• use \1 instead of $1 on the match side
• /g moves your start pos where the last match stopped, so you either need to use a lookahead (?=) or \K to keep the position after the first group
• your $1 will include whitespaces in the end and won't match at \n

here a demo with perl -dE0

 DB<70> $r = " #1 #2 #3 #1 #3 #2 #3 #2"

 DB<72> $_=$r; s/(#\S+\s?)(.+$1)(?{say "$_ ($1) - ($2) pos:",pos($_)})
+/$2/g ;say "fin: $_"
#1 #2 #3 #1 #3 #2 #3 #2 (#1 ) - (#2 #3 #1 #3 #2 #3 #2) pos:24
fin:  #2 #3 #1 #3 #2 #3 #2
[download]

you can use this as template to experiment and debug with (?{...})

update

like using \K

  DB<73> $_=$r; s/(#\S+\s?)\K(.+\1)(?{say "$_ ($1) - ($2) pos:",pos($_
+)})//g ;say                                             
#1 #2 #3 #1 #3 #2 #3 #2 (#1 ) - (#2 #3 #1 ) pos:13                    
+                                                       
#1 #2 #3 #1 #3 #2 #3 #2 (#3 ) - (#2 #3 ) pos:22                       
+                                                       
#1 #3 #2
[download]

update

like using (?=...)

  DB<81> sub DEB {say "$_ ($1) - ($2) pos:",pos($_)}

  DB<82> $_=$r; s/(#\S+\s?)(?=(.+\1))(?{DEB})//g ;say "fin: $_"       
+                                                        
#1 #2 #3 #1 #3 #2 #3 #2 (#1 ) - (#2 #3 #1 ) pos:4                     
+                                                       
#1 #2 #3 #1 #3 #2 #3 #2 (#2 ) - (#3 #1 #3 #2 ) pos:7                  
+                                                       
#1 #2 #3 #1 #3 #2 #3 #2 (#3 ) - (#1 #3 #2 #3 ) pos:10                 
+                                                       
#1 #2 #3 #1 #3 #2 #3 #2 (#3 ) - (#2 #3 ) pos:16                       
+                                                       
#1 #2 #3 #1 #3 #2 #3 #2 (#2) - ( #3 #2) pos:18                        
+                                                      
fin:  #1  #3 #2                                                       
+                                                                     
+                                                                     
+                                            

DB<83>
[download]

update

as you can see in the debug was \K not the best idea, but YMMV

update

you still need to fix the whitespace issue ...

... and IIRC there was a meta to keep the pos (\g or \G ?), please check perlre (I might misremember tho)