SSD Format for Windows and Mac

Yes, you can, but they can't be on shared space as is possible with APFS.

macOS requires the partition map to be GUID, as does Windows 10. For the drive partitions themselves, macOS requires APFS. When you use BootCamp Assistant to install Windows, it will set up a FAT32 partition which you then reformat as NTFS during the Windows install routine.

Once that's set up, use Disk Utility to create a separate hard partition to the size you want for the shared space and format it as ExFAT.

I recently bought a 512gb ssd and I wanted to split it into 256gb for mac and 256gb for windows. Is this possible with exFAT?

No. Neither macOS or Windows will install to an ExFAT partition. You'd need to split it into three separate pieces. APFS, NTFS and ExFAT.

ExFAT would be best as either platform can read/write that format without the need of any third party add-ons. But that would only be helpful for a separate partition. The Mac OS itself requires APFS, and Windows NTFS.

Is this SSD installed in a Mac internally or connected to the Mac externally via USBc/TB3/TB2/USB?

Here is an example of an internal disk with macOS (APFS) and Windows (NTFS).

 diskutil list 
/dev/disk0 (internal, physical):
   #:                       TYPE NAME                    SIZE       IDENTIFIER
   0:      GUID_partition_scheme                        *1.0 TB     disk0
   1:                        EFI EFI                     314.6 MB   disk0s1
   2:                 Apple_APFS Container disk1         616.0 GB   disk0s2
   3:       Microsoft Basic Data BOOTCAMP                384.2 GB   disk0s3

/dev/disk1 (synthesized):
   #:                       TYPE NAME                    SIZE       IDENTIFIER
   0:      APFS Container Scheme -                      +616.0 GB   disk1
                                 Physical Store disk0s2
   1:                APFS Volume Macintosh HD - Data     361.3 GB   disk1s1
   2:                APFS Volume Macintosh HD            11.3 GB    disk1s2
   3:                APFS Volume Preboot                 82.6 MB    disk1s3
   4:                APFS Volume Recovery                529.0 MB   disk1s4
   5:                APFS Volume VM                      14.0 GB    disk1s5

Bootcamp Assistant creates the necessary partition. You should not create any partitions manually, unless BCA has failed. Manual partition creation has rules, which Disk Utility does not follow. Creating 'Shared' partition is not officially supported, once Windows is installed.

From macOS Terminal, can you post the output of

diskutil list

Let it install and you can test. Once the installation is complete, post the output of

diskutil list

sudo fdisk /dev/disk0

You may need to temporarily disable SIP for the second command to work properly.

What is the year/model of your Mac (do not post the serial number)?

I assume you will replace the current disk in the main bay, and not add the SSD as a second disk in the Optibay. Is that correct?