Packages are being installed rather in proper directories/paths, not on disks as such - better to leave Windows thinking style aside.
What counts here is the directory tree - starting from / - aka root filesystem, and then, for example:
/boot
/bin
/sbin
/home
/var
/tmp
/root
etc.
Those can be just directories or mountpoints - where other disks can be mounted to.
From logical point of view then you'll see for example still the same /home, but instead of your SSD disk, it can be located on HDD disk.
You can install system on SSD disk and use the other, disk mounted under some mountpoint in your directory tree - for example /storage - to keep your stuff there. Whatever you prefer.
It doesn't even need to be the whole disk under /storage - depends on how you want to partition your disk.
Again - for example: /dev/sdb1 (first partition of your HDD disk, let's say) may be /storage, and /dev/sdb2 (the other partition - of whatever side) can be i.e. /home
If you have something on your hdd disk, that you'd like to keep - you, most likely, don't even need to format it - just specify the proper filesystem type in mount command.
Might be worth also to have a look at /etc/fstab if you want to mount the other disk automatically on startup.
Just anyway be sure also that you don't use as a mountpoint a folder with some stuff in it - after you mount another partition/disk to it, whatever was in the mountpoint directory will be inaccessible - until you unmount the disk again.
It may look a bit twisted, but it's not that difficult after all. But the topic is anyway a bit more complex than one could expect.
All depends on what you would exactly like to do.