It's not possible out of the box, because the umsdos driver which allows us to use Unix features on FAT drives were removed from Linux kernel 2.6
Umsdos is a linux file system. It provide an alternative to the EXT2 file-system. Its main goal is to achieve easier coexistence with Ms-DOS data by sharing the same partition. This document explain first how to use Umsdos in different configuration, and later explain its operation and try to provide some information letting you decide if it is a good choice for you (see UMSDOS-WHY-TO at the end).
https://tldp.org/HOWTO/UMSDOS-HOWTO.html
Without the ability to store permissions and symlinks lots of things will break, so no distro can offer that feature in their releases
umsdos
The key advantage to umsdos out of the three is that it provides full Unix file semantics. Therefore, it can be used in situations where it is desirable to install Linux on and run it from a FAT disk volume, which require such semantics to be available. However, Linux installed on and running from such a disk volume is slower than Linux installed on and running from a disk volume formatted with, for example, the ext2 filesystem format.[1][12] Further, unless a utility program is regularly run every time that one switches from running Windows to running Linux, certain changes made to files and directories on the disk by Windows will cause error messages about inaccessible files in Linux.
FAT filesystem and Linux
The modern replacement is POSIX Overlay Filesystem but since it's a FUSE driver, using it for rootfs would be tricky
A FUSE filesystem that provides POSIX functionality - UNIX-style permissions, ownership, special files - for filesystems that do not have such, e.g. vfat. It can be seen as a contemporary equivalent of the UMSDOS fs.