You can display only one additional field in Search Results. As per your screenshot, you have Account Name and you cannot have more than one field along with Name.
To customize: Go to Search layout of Parent Object (Opportunity) > Search Results > Edit > Add your field next to Name.
If you have multiple fields in Search Results, It takes first two acceptable data type fields in lookup results. (In your case, Opportunity Name and Account Name)
In this case, I have a Close Date in Search Results, So it will display on every Opportunity Lookups.
Please note, all type of fields are not supported in search results. For example, you cannot add picklist type fields in Search Results. Even if you try to add, it skips such kind of fields and display next availale field which is acceptable (Text, Date, Number & etc). ** I do not have acceptable types.
If the field that you want to see in lookup results is not acceptable, You can create a formula field and add it in 2nd position (Workaround). This way you can show that field in lookup results.
Also, if you want to display multiple fields (as per your question) you can separate them with comma (,) and add it in Search Results. - Workaround.
Your search also perform on second field. So please be careful
Hope it helps.