The button ahead of the e-mail address is a split button where the first entry is just, "Email." If you want subsequent addresses gain focus on the split part of that split button, expand it, pick Email 2, then you have a separate entry to put in the second email address. Lather, rinse, repeat for subsequent email addresses.
Note, if you wish to have the "Display As" for each email be different, e.g., "Carly Watson - Work," and, "Carly Watson - Personal," then be sure to edit the Display As field right after you enter the e-mail address for whatever email address entry number you're working on before you move on to the next one for that person.
A similar situation exists for physical addresses, too. That button is a split button and allows the entry of Home, Business, and Other addresses within a single contact.
Brian - Windows 10 Pro, 64-Bit, Version 2004, Build 19041
The purpose of education is not to validate ignorance but to overcome it.