This article explains how you can report on user searches for catalog items done in the service portal.
You can report on the sp_log table to view search activity with the service portal. For both search terms and the number of results, you can filter by Type = typeahead and Table is sc_cat_item.
There should be a count column which contains the number of results that were found based off of the value in the text column.
Also, we have a few OOB reports on this table that gather similar data which you can use for reference.