First, check if the file system of your Seagate SRD0SP0 1TB HDD is FAT, FAT32, or exFAT. Only the hard drive file system is FAT, FAT32, or exFAT, you can use the disk on both PC and Mac without formatting. Otherwise, the Mac Disk Utility will show this drive is 'Uninitialized' when the drive has a file system that macOS is not compatible with.
According to the error message that Disk Utility gave, there is a huge possibility that your Seagate HDD has a macOS-unsupported file system. Thus to make it usable on MacBook, you should format it with a macOS-compatible file system. Since reformatting will erase all data on it and the data is extremely important to you, you should recover files on it ASAP.
Actually, data recovery software is not so horrible. An unsafe data recovery app may bring viruses to your Mac indeed, however, a professional and safe data recovery app will save your precious data at ease. iBoysoft Data Recovery for Mac is professional and 100% safe data recovery software for Mac, which can recover data from external hard drives, SD cards, USB flash drives, memory cards, memory sticks, etc.
Even when the disk is corrupted, unreadable, unmountable, or unrecognizable, it can easily recover lost files in these cases. What's more, this program can recover lost data from unmountable macOS partitions, encrypted APFS boot volume, etc on macOS 13/12/11/10.15/10.14/10.13/10.12 and OS X 10.11.
Now, do the following to get files back from your uninitialized disk:
First, free download iBoysoft Data Recovery for Mac and launch it on your MacBook Pro.
Next, click on Storage Device from the sidebar. From the storage device list, choose your Seagate HDD.
Finally, click Search for Lost Data to scan all lost files on it. The lost files will be listed in this app, you can select all the files you need and click the 'Recover' button to recover data and store them on your MacBook or another healthy external drive to avoid data overwriting.
If your data has been recovered, you can format the external drive on Mac with a compatible file system to use it.