Why don't my photos automatically save on my MacBook Air? Both are Apple

What do you mean? That they don't get transferred when you connect your phone? If so, it was always necessary to manually import pictures to photos. If you are referring to a wireless/cloud transfer, you need to make sure you turn on iCloud photos.

Please read the following user tip and restate your question/problem in detail: Writing an Effective Apple Support Communities Question - Apple Community  

In the meantime:

• Which versions of Catalina and Photos are you running?
• Is your library a managed or referenced library?

• Where is your library located?  If on an external HD how is that drive formatted and connected to your Mac?
• Do you have the iCloud Library enabled in both devices? 
• If you're not using iCloud Library do you have My Photo Stream enabled on both devices?

An remember, just because they are both Apple products you still have to set them up to do the things you want them to do.