Users see different auto-suggested results when a Search is performed in Lightning Experience

This is Working As Designed. In Lightning Experience, there's no way to specify either Standard or Enhanced Lookups, both lookups are served by search as long as they are searchable in the case of custom entities. The behavior also holds true for Global Search in Lightning Experience. 

For more on Lookups and their differences, please review the following blog post  

This is also documented as follows:

"Recent items only show the record’s object type; other matches also show the secondary field. Selecting an item takes you to its record page."
                                                                                                             
" If the list of records in instant results includes recent items and other suggested records, users see a mix of results with and without secondary fields."
 

Here's an explanation of what happens step by step:

1. When you first click (or focus) on a lookup field, the system will identify 25 of the users' Most Recently Used [MRU] records and display the top 5 records. 

Note:

• Most Recently Used [MRU] records are those that the user has recently accessed; they are similar to what the user may see under the Recently Viewed List View for the object in question.
• The MRU takes some time to update for the user, so if you do not see the desired results, please check back after some time.

2. When you type the first search character (the number '1' based on the example above), the results will filter those same 25 MRU records to display only the records that contain that character. 
3. The same behavior occurs when you type the second character (the numbers '12' based on the example above), search will check the original 25 records to display only the records that contain both of those characters.
4. On the 3rd key stroke, behavior in the Lookup field changes from displaying the MRU records to type-ahead search. Type-Ahead search works differently in that it uses other variables such as stemming, spell-check, and related records. 

Due to the difference between MRU and type-ahead, it is possible to see records appear in the results for 1 or 2 characters but not in the results for 3 or more characters. 

It is also possible that 2 users will see different results in a lookup field, while it depends on MRU. Once they start typing 3 or more characters, they start to see more similar results since it is no longer based on MRU. 

Based on the above, the resolution workaround here would be to:

1. Have the affected user Access the records prior to the lookup at some point; this would put these records in the user's MRU
2. Use actual search terms to see desired results. The more restrictive the search term, the better the user's chance of seeing desired results.