I don't think it's possible... even in the pgAdmin III UI, if you try to edit a constraint created with your statement, the "Unique" box is greyed-out; you can't change it through the UI. Combined with your research on the ALTER INDEX docs, I'd say it can't be done.
If you have PgAdmin4 installed than it's very easy just follow below steps...
Go to table properties(right click on table and select properties).
A window will open than navigate to Constraints in properties window.
In Constraints option you will see options as below screenshot just go to Unique there you will see columns that has unique constraints. Click on delete button and save. that's it you are good to go.