This is because the new Email Builder sends emails as multipart/alternative MIME-encoded with a plain text section containing the content you mentioned, as well as the HTML section which has the body of your email template. The plain text section is placed at the beginning of the email, and some mail clients will choose to display this text as the “second subject line”.
To the best of my knowledge (I work more on backend systems and don’t use the Infusionsoft interface every day), there is no way to change this in the new Email Builder. The plain text portion isn’t available for editing and there isn’t an option to send the email as plain text only or HTML only. The legacy email builder does have the option to send as plain text only or HTML only, but I don’t know offhand whether you can modify the plain text content or reorder it to be after the HTML content.
Making this “second subject line” more customizable is something I’ve suggested to our developers recently as a feature request, and I’ll pass along your feedback as well.