This might depend on how your Cockpit instance is setup - “Lite” or “Db”.
I’m new to the CMS as well, so I’m only dabbling on what I can read from the source.
A) MongoLite
As far as I can tell there is no build-in option to use the LIKE operator.
But for the MongoLite db driver there is a filter option using regular expressions $regex that you might be able to utilize instead.
It is implemented in cockpit/Database.php at 568b0124352f6d27df359e8c19a70d2dd1961e87 · agentejo/cockpit · GitHub
B) MongoDB
The MongoDb driver uses the collection->find($filter) method.
While it also supports the $regex filter option, it might allow even more.
Have a look at https://docs.mongodb.com/manual/tutorial/query-documents/ to see all the different options on how to mold the $filter object in order to maybe achieve a LIKE request.
Though when searching the MongoDb docs it seems as if there is no LIKE operator but that $regex is actually the operator of choice for such a request.