Just use awk:

$ awk '!($NF in users){ users[$NF]="user"(++cnt) } { $NF=users[$NF] } 1' file
Name 1:10:34 date short_id user1
Name 1:10:45 date short_id user1
Name 1:20:54 date short_id user2
Name 1:30:43 date short_id user3
Name 1:40:43 date short_id user4
Name 1:40:13 date short_id user4
Name 1:20:01 date short_id user2
Name 1:10:01 date short_id user1

Just use awk:

$ awk '!($NF in users){ users[$NF]="user"(++cnt) } { $NF=users[$NF] } 1' file
Name 1:10:34 date short_id user1
Name 1:10:45 date short_id user1
Name 1:20:54 date short_id user2
Name 1:30:43 date short_id user3
Name 1:40:43 date short_id user4
Name 1:40:13 date short_id user4
Name 1:20:01 date short_id user2
Name 1:10:01 date short_id user1

The above assumes you want each user to get a unique ID based on the order they appear in the input. If that's not what you need then edit your question to clarify your requirements.