While SilverFace and McKay both have excellent answers, I feel part of the story has been left out.
The UK, and by extension most if not all current and former Crown colonies, inherited their use of the first name as a manifestation of close interpersonal ties from French and German, wherein the distinction between tu and vous (French), or du and Sie (German), fills the same societal role as the use of first names. In fact, the French tutoie-moi (verb tutoyer), generally translated as be friendly with me literally translates as something like 'tu' me. German has a similar construct with duze mich (verb duzen), which while sharing its literal meaning with French ('du' me), is translated by Wiktionary as either to address informally using the pronoun du, to thou, or to be on first-name terms.
Why is this relevant? Because the further you go from the source of this usage, the more it has drifted to a form of overformality. In the UK, being on first-name terms has some meaning, as most people over a certain age (somewhere around 35, though I can't find satisfactory evidence it's been studied) tend to refer to others with either given names or surnames depending on status, with given names used for those of equal or lower status, and familiarity, with the notable exception of royalty often being referred to using their proper honorific (eg. "Sir", "Lord", "Queen", and "Prince") and their given name.
In the US and Australia, this is largely no longer the case once one reaches the age of majority. American schoolchildren are expected to refer to teachers by their surnames, and in some areas all adults; Australian and Canadian schoolchildren similarly use the surname for teachers, though this may be starting to change. In adulthood, with the exception of doctors, political figures (with some exceptions), and faculty and staff within higher education (professors, deans, etc.), one generally refers to others by their first name only.
To answer the question of what it is called in American English when two people decide to call each other by their first name, I would say normal. Two people mutually referring to each other by surname would be considered very unusual, as essentially all remaining surname-only relationships are unilateral, i.e. teachers refer to students with their first names, and doctors similarly refer to each other by first name. Perhaps more tellingly would be if you referred to two people as being on a last-name basis, which as mentioned by SilverFace would be far more commonly said in the US than being on last-name terms.
Finally, to ask someone to refer to you by first name in the US, if they aren't already, I'd try "Call me Bob."