Why is the inode cache needed

Then why do we even need a directory cache?

It's just another level of caching. Even if all the path components can be found in other caches, it still takes time to traverse them all. So the directory cache provides a way to cache whole paths and avoid looking things up path element by path element from the root each time.

They're on different devices.

If we look at the output of stat, we can also see the device the file is on:

# stat / | grep Inode
Device: 801h/2049d      Inode: 2           Links: 24
# stat /opt | grep Inode
Device: 803h/2051d      Inode: 2           Links: 5

So those two are on separate devices/filesystems. Inode numbers are only unique within a filesystem so there is nothing unusual here. On ext2/3/4 inode 2 is also always the root directory, so we know they are the roots of their respective filesystems.

The combination of device number + inode is likely to be unique over the whole system. (There are filesystems that don't have inodes in the traditional sense, but I think they still have to fake some sort of a unique identifier in their place anyway.)

The device numbers there appear to be the same as those shown on the device nodes, so /dev/sda1 holds the filesystem where / is on:

# ls -l /dev/sda1
brw-rw---- 1 root disk 8, 1 Sep 21 10:45 /dev/sda1