In our area, single family homes are mostly fed with 240 volts, while many apartments and condos are fed with 208 volts. Yes, you can have the electrician install a NEMA 14-50 receptacle that you can use to plug in the Tesla Mobile connector. If you have the Model 3 Long-range battery, while the Mobile Connector will charge fine, it won't charge the Model 3 at the maximum rate (32 amps vs. 48 amps). If you have a mid-range Model 3 battery, then that is a moot point. If you don't drive that many miles a day, then the charging speed may not matter much to you anyways.
You can choose to use the mobile connector that came with the car in the garage to charge. I would not recommend unplugging it and putting it back in the trunk every day, as the receptacles aren't meant for that many plug/unplug cycles. If you want something in the trunk to charge for times away from home, then you can buy another mobile connector.
Or, alternatively, you can buy a Tesla wall connector for about $500 new plus tax from Tesla and have that installed on the wall in your garage (and leave the mobile connector in the trunk for those times when you need it out of the house - usually not often). They make a hard-wired version of the wall connector and a version with a short pigtail that would plug into a NEMA 14-50 receptacle.
The electrician will need to verify that you have physical space for the new circuit in your panel, and also should do some load calculations that are specified in the National Electrical code to ensure that your electric service has the excess capacity to handle the new charging circuit. Permits are required in most jurisdictions, but it's up to you (or the electrician) to get one. When I sold a house a few years here in California, I had to sign a disclosure that I didn't have any unpermitted electrical work. Since you're a tenant, the homeowner may want to get the permit. That would be between you and the homeowner.
What other questions do you have?