In Gmail, my email address is replaced by "me" in emails and chats, etc.
Is there a way to replace this "me" by anything else, such as the actual name of the person?
Gmail will always show "me" for your Gmail address.
If you are using "send mail as" addresses, untick "Treat as alias" and we should stop showing "me" for those addresses; this is the preferred option for addresses that you send mail as but do not reflect your personal identity, such as mailing lists.
If you are seeing a problem with senders who have firstname.lastname@example.org as their email address, you can correct that issue by updating their information in your Gmail Contacts.