I have covered “mate” a couple of times as a synonym for “buddy” or “friend.” But this sign in a men’s room in the Seattle airport is the first time I’ve seen it in America as a form of direct address:
The capital M in “Mate” suggests to me that the writer wasn’t especially comfortable or familiar with the term. Meanwhile, both the excessive politeness and the “eh” at the end suggest that he or she might be Canadian.
In any case, the first two citations in the OED for this use of the word are both from Englishmen (Arthur Polehampton and Lord Robert Cecil) who noticed it in their travels in Australia in the mid-nineteenth century.
1852 R. CecilDiary 31 Mar. (1935) 36 When the diggers address a policeman in uniform they always call him ‘Sir’, but they always address a fellow in a blue shirt with a carbine as ‘Mate’.1862 A. PolehamptonKangaroo Land 99 A man, who greeted me after the fashion of the Bush, with a ‘Good day, mate’.
It had arrived in Britain by 1880, when this line of dialogue appears in a novel by Mary Elizabeth Braddon: “Who’s the magistrate hereabouts, mate?”
In the U.S., the comparable terms include “buddy,” “pal,” and, in recent years, “dude.” Those words are often used in a hostile, or at least passive-aggressive manner. “Mate” works well for this purpose, as the men’s room admonition illustrates. I’ll be curious to see if it catches on in these parts.
notoneoffbritishisms.com 
So did u put the seat up or no, matey?
This Safire piece from ’84 identifies direct address forms such as “buddy” and “mac” as Northeastern: https://www.nytimes.com/1984/06/17/magazine/on-language-my-name-ain-t-mac-buddy.html
“Dude” was at that time an up-and-coming challenger to those old-school forms.
I wonder whether “It’s mate against mate, state against state” will ever be used as an advertising tagline in the US? I first saw this as a slogan for the annual State of Origin Queensland vs New South Wales men’s rugby league series in Australia – which I used to follow closely from the other side of the Tasman Sea – but a quick search reveals that it’s now used to refer to other Australian sports and politics as well.
“Don’t come the raw prawn with me, mate” may be a little too specialised for US tastes.
Just a guess here but I’d imagine this may have been the first time Messrs Polehampton and Cecil had actually met and conversed with members of the working classes and exposed to their speech.
Lots of slang that is claimed by Australians these days originated in the UK but is now considered archaic (dags, larrikin, etc) in the mother country.
There are pictures of copies of the sign in different places, including a “Canadian ferry bathroom”: https://imgur.com/Nw8kKIm. I wonder where it originates from.
Great find! I posted the picture on Twitter and immediately heard from the Seatac Airport Twitter person, who said that it wasn’t official, and looked like someone had put it there. BTW, the sign has been copyedited (subedited) since 2018, with the word “be” inserted.
‘Mate’ can also be used in a hostile passive-aggressive manner. And in about 25 other ways.
Traditional reference joke: Australia, where you call your mate “c**t” and call a c**t “Mate”
‘Mate’ is in common usage in Britain as an affectionate form of address between men, though not, I think, among the privileged classes. It can be a term of menace, as Americans might use ‘pal’.
In today’s Guardian there is a letter from some who found, in Screwfix in Oxford “In the space of a couple of minutes on Saturday I was addressed as fella, chap, guy, buddy and mate by just one counter assistant.”
https://www.theguardian.com/politics/2019/jul/29/missing-the-churchgoing-theresa-may-already
”China”, the cockney rhyming slang version of ”mate”, is also fun to use. It can however cause confusion to those unfamiliar with that argot.
I did not know “buddy,” “pal,” and “dude.” are often used in a hostile or passive aggressive way. Very enlightening.
You use the term ‘room-mate’, so mate is not completely alien in the US.
I’ve always thought American English “man” (as in “Thanks, man,”) mapped rather well, if imperfectly, to Australian “mate”. Particularly in the case of the posted photo.