Since email is used for authentication, how would you want the system to consider you as a user without keeping the email ? Come on, be serious for 2 minutes.
No, you don't have to. You want to.
The concept of tying email addresses to user accounts came Google to better track their users. Of course thankfully taken over from everybody in the industry.
Especially when you offer a "free" trial for your software you should not ask your users to pay with their data. Even first before they see what they'll get for in exchange.
You literally ask me to pay the price first before I see the product
Easy.
I would be surprised if your app would have a "change your email/login" function implemented.