Seagate external drive is read/write on one MAC and read only on new MAC

macOS can only natively read NTFS (Windows) volumes.

You have to install software to write to NTFS (Paragon, Tuxera).

Drive manufacturers bundle absolute crapware versions of those products with the drives instead of explaining how to format the drive for use on a Mac. You likely installed that on your other Mac.

Unless you need to use that drive with Windows and it must have permissions, there is no need to use NTFS.

Copy all of the data off that drive and use Disk Utility to Erase it as macOS Extended (Journaled). Then, copy the files back onto it.

In Disk Utility, you'll have to select "Show All Devices" from the view popup menu. It will then show the actual drive device in the list. Select that and Erase it, not the "4 Terabyte Blue" volume.

FWIW, macOS doesn't support writing to an NTFS drive without a driver. You can get a free driver from Seagate.

https://www.seagate.com/support/software/paragon/

If it was working with another Mac then you were running a third party app such as Paragon or Tuxera that enabled writing to a NTFS formatted drive.