Can I continue my save on a new switch with a new game copy

So my dad has a switch which I managed to briefly use to play breath of the wild. However, recently he's decided to never let the switch out of sight so I'm thinking of saving up for my own console and copy of the game.

The problem is that I want to continue with my progress that I made on his copy. Is there any way that I can get my save file onto a new switch with a new cartridge for the game?

Yes you can. You will need both Switches.

From the Nintendo Help pages:

On the source console:

1. From the HOME Menu, select "System Settings" > "Users" > "Transfer Your User and Save Data."
2. Select "Next," and then "Next" again, then select "Source Console" to indicate that the content will be transferred from this console.
3. Select "Continue," and then finish the following preparations on the target console.

On the target console:

4. From the HOME Menu, select "System Settings" > "Users" > "Transfer Your User and Save Data."
5. Select "Next," then "Next" again, then select "Target Console" to indicate that the content will be transferred to this console.
6. Select "Sign in" and choose to sign in using your Nintendo Account e-mail address or sign-in ID.
7. Enter either the e-mail address or sign-in ID associated with your Nintendo Account, and the password, then select "Sign in" > "Next."

On the source console:

8. Wait for the target console to be found, and then select "Transfer."
9. The transfer process will begin, wait for this process to complete.
10. Once complete, select "End" to complete the process.
    • If no users are left on the console after the transfer is complete, a placeholder user account will be created.

They also include a helpful video on how to do this:

As for if you can play different saves on different copies of the game, yes you can.

As of September 2018, saves can be backed up to the cloud, and restored without the original Switch. However, you must pay for online service to use this feature, and you must have purchased online service before losing access to your old Switch.

According to Nintendo's product page, backups are automatically taken whenever you have an internet connection (and you have paid for online service). Their help page describes the following procedure to download cloud saves onto a new console:

• To download a backup from the software menu:
  1. Open the game's software menu and select "Save Data Cloud Backup."
  2. Select the desired user, then select "Download Save Data."
• To download a backup from within System Settings:
  1. From the HOME Menu, select "System Settings" > "Data Management" > "Save Data Cloud Backup."
  2. Select the desired user, then choose the backup file you wish to download.
  3. Select “Download Save Data.”

The page also warns that this procedure will erase any local saves. The same settings page can also be used to verify that your saves have backed up successfully, if for some reason you're concerned about that.