Can you give more details about your problem?

Can't you connect to your Sync account?

If you connected your current computer (device) under a different name then can might still see the previous computer under its original name.
You shouldn't worry it you see this device present (e.g. under tabs from other deices).

Note that you need the account name (email) and the password and the Sync key to connect successfully.
If you didn't backup the Sync key then you need to generate a new Sync key and you lose all data stored on the Sync server.

• https://support.mozilla.org/kb/how-do-i-set-up-firefox-sync
• https://support.mozilla.org/kb/ive-lost-my-firefox-sync-account-information

My issue is exactly:

"If you connected your current computer (device) under a different name then can might still see the previous computer under its original name. You shouldn't worry it you see this device present (e.g. under tabs from other deices)."

As my other mobile device show also the "old device", it looks odd and my wife confuses (we share the same firefox sync).

Can anyone know how to delete the 'old device'? Thanks a lot.

I think that the old device will go away automatically, but I do not know how long his will take because you didn't unlink that old device.
You could try to connect to that account and use the old name and then unlink this device properly or you can generate a new Sync key to wipe all data on the server and reconnect all devices with the new Sync key.