Don't let someone with a suspended license drive your car. As the owner, you are supposed to make sure anyone who drives your car has a valid license. If you don't, you risk having your car towed.
If he was legally arrested, they could legally tow the car and impound it for up to thirty days. See California Vehicle Code §14602.6.
If the car was legally impounded, then they could search the car to 1) document any valuable items that were in it, so you couldn't later claim someone took your $5000 Rolex watch from the glove box, and 2) remove any contraband.
Note that I have some questions on whether the arrest was legal if the officers didn't actually see him driving it, but that is a complicated legal issue that would require a full review of all the circumstances.
Please understand that this is a general discussion of legal principles by a California lawyer and does not create an attorney/client relationship. It's impossible to give detailed, accurate advice based on a few sentences on a website (and you shouldn't provide too much specific information about your legal matter on a public forum like this site, anyway). You should always seek advice from an attorney licensed in your jurisdiction who can give you an informed opinion after reviewing all of the relevant information.
Based on what you wrote, there may well be a valid motion to suppress the evidence. They would have to justify the search under some grounds, but if they didn't impound it or tow it, there's no legal basis for an inventory search.
If your boyfriend was arrested, they could search the passenger compartment pursuant to his arrest, but as my colleague mentioned, without evidence that he drove, the arrest appears suspect.
Your boyfriend, if he hasn't already, needs to consult with a criminal defense attorney regarding this situation.