No. The free trial and the welcome Flares are once per person. A new account still works, but it starts on the free tier without them in three cases: its email address already had them on an account that was later deleted, the device has already had them on another account, or the address is a temporary or anonymous mailbox.
Nothing else is different for those accounts — the free tier works as normal and a subscription can be started at any time. If you signed up with a temporary mailbox and would like the trial, sign up with Google or with a regular mailbox instead. If you think an account was caught by mistake, for example on a shared computer, contact support and it will be looked at by a person.
To make this work, two scrambled one-way markers are kept for twelve months and used for nothing else: one of the email address of a deleted account, and one of a random identifier stored in your browser when you sign in. Neither can be turned back into your address or identify your device, and the Privacy Policy describes both.