Fax a document with text message

No. SMS Text is a Cellular messaging feature essentially limited to ASCII text; SMS doesn’t allow you to send images or attachments. MMS Messaging, this being another Cellular feature, allows sending of full-colour images in addition to text.

FAX is a much older technology that sent a monochrome “scanned image” of a page - the page content (text, images, or both) being totally immaterial.

No, completely different technologies. As long as the document is received in a printable format, the end result is effectively the same. So would emailing the document as an attachment.