I don't believe so. Someone please correct me if I'm mistaken about any of this, but the FAT boot partition you're referring to is not the same thing as the partition upon which the /boot directory is mounted. Rather, it's just a small partition, usually about 512 KB in size, which the BIOS or UEFI firmware uses as a workspace during the boot process. When I create these partitions during my BIOS installations, I specify the type of this partition as "freebsd-boot" rather than as "freebsd-ufs" or as "freebsd-swap" which are the only other types I use regularly. It's been too long since I did a UEFI installation but there the partition type might have been specified as EFI like it is on Linux systems... I just can't remember what it was called.
The partition type of the /boot directory must be a journalized unix filesystem type like ufs, zfs, or ext which supports unix-style user and group ownerships, and permission bits. FAT file system types don't support ownership or permissions for individual files and directories. When you mount an msdosfs drive under FreeBSD or other unixes, the system just sort of fakes the ownerships and permission bits, by copying the user and group ownership of the mount directory, and by pretending that the files and subdirectories under that directory all have the same ownerships as the mountpoint directory has, and by applying some sort of common permission scheme to all the files and directories under the mount point. Usually it's something like 755 for directories and 644 for flat files. Depends on the mounting software used, and on how it's configured.