Of course, people can give fake names and email addresses (unless confirmation is enabled) which effectively makes them anonymous.
However this might compromise the integrity of the petition giving the target an excuse to ignore it.
It is possible to have a signer’s real details sent with the petition, but hidden completely on the public signature list.
Edit your petition and in the right hand column notice the option to “Allow anonymous”. Enabling that will display “anonymous” in the list of signatures if the signer chooses the “hide name from public” option.
In all other aspects, the name is treated as entered.