Rattledagger

I've never tried this particular Vortex extension, but you can in Vortex under "Extensions"-tab click "Find more" and search for "Download Game Assigner".
 
After installing this extension and re-starting Vortex, under Settings - Downloads where should now be a new option where you can assign downloads for game X is compatible with game Y and afterwards any new downloads for game X should be possible to install in game Y without getting any warning about "different game".

Some games only works with hardlink deployment and hardlinks only works if game and Vortex mod staging folder is on the same disk. Example, the Fallout games and the Elder Scroll games only works with hardlink deployment.
 
Meaning, depending on game, it's maybe not possible to move mod staging folder to HD, unless you also move the game to HD.
 
In any case, to move mod staging folder, you must do this inside Vortex, by specifying a new location of the mod staging folder and Vortex will move the files for you.

ctrl+a, choose "Remove" to uninstall all mods. If you also wants to delete the mod-archives choose "Delete archive" in the dialogue that opens-up after choosing Remove on all mods.
 
To be even more extreme, you can also delete all save-games and delete all profiles.

It's important to let Vortex move the "Mod Staging Folder", since if you do this manually it's a good chance all installed mods will be duplicated and you'll lose all the conflict-rules.
 
So basically, you should do something like this, where let's say you first select SSE:
1: On Vortex mods-tab, click "Purge" and afterwards exit Vortex.
2: In Steam, move game to another disk.
3: Run game launcher to setup registry etc.
4: Start Vortex, you'll get a message about game moved.
5: Just to be on the safe side, re-start Vortex.
6: Make sure new location of game is detected in Vortex.
7: Under Vortex - Settings - Mods, type-in a new location of "Mod Staging Folder" to be on the new disk.
8: Let Vortex move all mods for you, this can take some time.
9: Under Vortex - Settings - Mods, make sure you're using "Hardlink Deployment".
10: Under Vortex mods-tab, click "Deploy".
11: If you want to move another game, in Vortex switch to another game and start on step 1 again.
 
For step 1, the main reason to use Purge is to speed-up step 2, since if you don't purge the mods you'll copy all active mod-files twice, first in step 2 and afterwards in step 8. A minor reason is, it's possible some of the files isn't correctly handled afterwards, meaning even if you disable or uninstall mod the files will be left behind.
 
It's possible you can move multiple games at once, but it's safer to take only one game at a time.
 
Note, if you Purge and switch to another game, Vortex will re-deploy game before switching to another game, making the Purge a waste of time.

Then it comes to Morrowind, you'll need to carefully read the installation-instructions for each mod to find out if you can directly install the mod with Vortex or not. Any mod that says something like "use NMM to install" you can use Vortex to install.
 
While many Morrowind-mods can't be installed directly, it's still normally possible to use Vortex to extract the mod-archive for so later use Vortex to manage the mod. To extract the mod-archive, choose an uninstalled mod on Vortex mods-tab, righ-click mod and choose "Unpack (as is)". Afterwards, right-click and choose "Open in File Manager". In File Explorer that opens-up, here you'll need to move the files/directories around to follow the installation-instructions. If a mod should be installed to the main Morrowind-directory, on Vortex mod-tab double-click the installed mod to open-up the extra dialogue on the right and for "Mod Type" choose "dinput". At the end, Enable mod, handle conflicts, Deploy and handle plugin load order.

Uhm, do you mean the "Guess IDs" option you'll see in the new menu opening-up near the bottom of the screen if you select multiple mods in Vortex mods-tab? If so, feature is already implemented...

The default deployment-method with Fallout4 is hard-linking, this means the Mod Staging Folder must be on the same disk as the game. So, I would do something like this:
 
1: In Vortex on mods-tab for Fallout4, hit "Purge" and then finished exit Vortex.
2: Use Steam to move Fallout4 to the new disk.
3: Run Fallout4 launcher once, to setup new registry location etc.
4: Re-start Vortex, you'll get a message about Fallout4. Make sure Vortex has detected the new location of Fallout4.
5: Just to be on the safe side, re-start Vortex.
6: In Vortex under Settings, Mods, change "Mod Staging Folder" to be on the new disk. Vortex will now move all mods to the new disk, this can take some time.
7: Make sure under Vortex, Settings, Mods, you're using "Hardlink Deployment".
8: On mods-tab, hit Deploy.
 
Any short-cuts to Fallout4 in Vortex Dashboard may possibly also need to be updated.

Where are basically 4 ways of doing this:
 
1: Re-download everything and re-install everything.
2: Copy the downloads over and re-install everything.
3: Copy downloads + mods + %appdata%\vortex + anything directly installed to game directory.
4: Copy full game directory.
 
1 is definitely the most boring and time-consuming, since even if you've got a direct link to Nexus download-page it can still take many hours of downloading all the mods. It's also very easy to make a mistake, some mods can have been permanently removed from Nexus while some mods isn't available on Nexus and can also be difficult to download. In addition you'll need to re-create all rules etc.
 
2 is much faster than 1 since you don't need to do the very slow re-downloading of everything. Re-installing everything is also fairly fast, but you do need to remember all the FOMOD installer choices. You'll also need to re-create all rules etc.
 
3 is the largest amount of files to copy, but apart for transferring of the files you'll only need a couple minutes of configuration to be up and playing. Since you transfer "everything", you'll have a fully moddable setup there you can easily disable some of the mods if don't like them, and add new mods on top.
 
4 is somewhere in the middle between 2 and 3 then it comes to amount of files that needs to be transferred. Since you transfers the full Skyrim-directory after transfer finished you'll be ready to play the game, meaning time-wise this is the fastest option. The big disadvantage here is it's nearly impossible to disable a mod, except if mod only comes as esp + bsa, and adding more mods isn't a good idea either.
 
To me 3 is the best, since this gives a fully moddable setup where it's easy to remove or add more mods and it's also an excellent starting-point if wants to mod more. Alternatively 4 can be used, in case you expect the modden game will only be used for a single play-through where no mods will be removed or added and don't expect will do any more modding in the future.
 
The only major weakness with 3 is, downloads + mods must be located at exactly the same place on both computers. This can be a problem if you've example got mods on E while the other computer only has a C. The game itself doesn't need to be at the same location. In case game isn't on the same partition you'll also need to either move the game or the mods folder, but this can be done in either Steam or Vortex so isn't a big problem (apart for using longer time).
 
As for the actual transfer of the files, as long as you've already got an USB, even if this USB isn't large enough to copy everything over in a single instance, by using 7-zip you can easily create archives with a specific size. Example a quick test revealed it took me roughly 7 minutes to create 44 pieces each of 1000 MB in size out of my Skyrim mods-directory. If you've example have got a 16 GB USB this means you can transfer these roughly 43 GB of mods by making 3 trips to the other computer. Note, in this test I used the option to "store" the files, meaning I didn't waste some hours trying to compress the mostly uncompressible bsa-archives etc.

Game and mod-folder must be on the same drive, so you should do something like:
 
1: In Vortex, on mods-tab hit "purge".
2: Exit Vortex.
3: In Steam, move Skyrim to another drive.
4: Run Skyrim launcher once to setup registry etc.
5: Re-start Vortex to detect new location of Skyrim.
6: Just to be on the safe side re-start Vortex.
7: In Vortex click Settings, Mods, specify new location of "Mod Staging Folder" on the new drive, and Vortex should move all files for you. Note, this can take a long time.
8: On mods tab, Deploy.