Older Game: New Mac Computer

There are two issues in play here.

The first one is that no macOS from El Capital (10.11) or higher will authenticate from the retail discs anymore. The game has to be installed by way of Origin for Mac, which it can be at no extra charge if you are the original owner of the discs and they are or can be registered to your account.

The second issue unfortunately cannot currently be overcome. TS3 is a 32-bit game only and Apple dropped support for all 32-bit programs with the release of Catalina (macOS 10.15). The game will just not install nor run on that system. A fix is in the works for this as EA plans to release a compatibility patch later this year that will allow the game to run on Catalina. The only way to make that happen in the meantime is to have access to a drive/volume that will boot an earlier macOS like Mojave or High Sierra, or Bootcamp Windows onto your Mac and install/play TS3 for Windows on it instead if you have the drive space and patience to devote to managing two operating systems and are willing to purchase a retail standalone version of Windows 10 to install.

Edit: Sorry Puzzlez, didn't realize you were moving this post as I was responding to it but you're going to have to type about 31 seconds faster (sometimes).    :D

@k_ker21  I've moved your post to the Mac section.

The disc issue isn't new: all versions of macOS starting with El Capitan (10.11) will fail to authenticate Sims 3 discs.  But El Capitan is newer than the game, so that wouldn't have made it into the official requirements printed on your disc case.

The more immediate problem though is that Sims 3 doesn't run in Catalina (10.15) at all.  The game is a 32-bit app, and Catalina is 64-bit only.  There's no way around this issue, at least for now.  EA has announced a 64-bit-compatible version of the game, to be released (hopefully) this year, but it's not clear at all when this is going to happen.

https://answers.ea.com/t5/Technical-Issues-Mac/An-announcement-for-The-Sims-3-players-on-Mac/td-p/8271641

In the interim, your only options are to play on a different computer, to install Windows alongside macOS (via Bootcampe or Parallels), or to install Mojave on an external drive or USB and boot into that when you want to play.  For the second option, you'd need enough enternal storage to dedicate to Windows: 100 GB is probably the bare minimum, plus enough left over for the Mac side.  For the third option, you'd need temporary access to a Mac running Mojave or earlier.  Let me know if you want more information about any of these options.

Edit:  Yes @igazor, I'm moving a bit slow today.  I take 16 hours off from posting, and suddenly my fingers barely work anymore.