I want to keep track on what a customer searches in my portal. Does Salesforce have a standard object able to implement this requirement?
The only standard feature I can think of that might solve this is a debug log that captures the debug logs from the class. You would have to set up a debug log in a date time range and it would only stay round for 7 days. You can read up more and debug logs at this link: https://help.salesforce.com/articleView?id=code_monitoring_debug_logs.htm&type=5
However this would capture a lot of activity and not just what search text was used. For something like that I would suggest creating a custom object that captures that information every time a search string is entered and used in the SOSL search. You would need to update your code so that a new record is created every time a search string is entered.
I would advise against something like that or dumping this into an external database and periodically purging the records from your system. I would suggest this because you will quickly run out of space in your org if you are constantly creating records every time a user does a search.