Register with the police within 3 days?

You need to check with your Airbnb host - they should do this for you.

https://www.gov.uk/foreign-travel-advice/slovenia/local-laws-and-customs

In Croatia, the accommodation owner has to take your passport details to the authorities within a day of you arriving. It’s a hangover from the old communist days. I presume it’s the same in Slovenia.

It's not just a former communist thing...many European countries have varying requirements similar to this.

In practice it is never checked. The accommodation owners may be checked, but not indivuduals.

I think if you tried to go to a police station and say that you wish to register they'll probably laugh at you.

I crossed the borders of Slovenia dozens of times(car, train, on foot) between 1993 and 2009 and they never asked for registration.

These days they don't even stop you at the borders(except the Croatian) because they're in the Schengen Area. If a police officer stops you he'll be interested in the documents of your car, perhaps your driving licence and your ID(passport), nothing else.

Hello, my boyfriend and I are planning on going to Slovenia for two weeks this year. We are both nineteen years old and will rent a car in Trieste, Italy (minimum age for car rental in Slovenia is 21).

I'm not sure this is correct. I did see that noted on a few websites, but car rental agencies seem to allow it.

This site says you need to be 18 but have had "at least 3 years of driving experience."

https://www.ljubljana.info/car-rental/

Avantcar (rented from them two years ago in Ljubljana - I'm over 21) will rent to people under 21 (for an additional fee) who have had their driver's licenses for two years:

http://www.avantcar.si/en/general/more-information-2/frequently-asked-questions/terms-and-conditions-of-rental/#verticalTab1

Auto Europe (broker) claims you can rent a car in Slovenia if you are over 18 and have had a driver's license for one year:

https://www.autoeurope.com/driving-information/slovenia/

Numerous countries around the world require tourists to be registered and it is usually the onus of the hotel or AirBnB host. This is why people are asked to provide their passport on registration. They might take a photocopy of it so they have the details to register you without holding everyone up. I have had AirBnB hosts request a scan of my passport prior to arrival - insisting on it being a requirement for rental. That is fine with me. Others want you to provide a photocopy of it. So it doesn't hurt to have a couple on hand for such purposes.
In North America we tend to have this fixation about letting others see our passport or letting it out of our hands, but the reason to protect it when travelling is really because it can be a hassle to replace it on the road. It is really our only valid identification when traveling as we don't possess that country's official ID card. A driver's license is not recognized as 'official' ID.