Using "me" (or indeed other object pronouns) like this generally considered to be grammatically incorrect, because a subject pronoun ("I") should be used as subject of the verb.
In formal (and probably everyday middle-of-the-road) language, this misuse of object pronouns is to be avoided. You should use "Adam and I", with "I" after the "and", because "I and Adam" sounds awkward (and grandiose).
But "me" as a subject is relatively common in informal language. Certainly I remember using sentences like "Me and Simon are going to the park" in the school playground, only to be rebuked by my teachers in the classroom.
In the classroom we learn to say "Simon and I are going to the park" instead, using a subject pronoun, because Simon and I are the subjects of the verb to go.
If we have good teachers, we learn to check by removing the other person: "Simon and I are going..." becomes "I am going", which sounds ok, so we say it. This way, when the pronoun should correctly be an object pronoun, we get it right even when Simon is in the way: "The dog attacked me" so "The dog attacked Simon and me". And we don't have to learn nasty words like accusative.
If we didn't pay enough attention though, or had bad teachers, we'll think the rule is that you never say "Simon and me" and always "Simon and I". So we start producing hyper-correct forms like "The dog attacked Simon and I", where "I" is a subject pronoun instead of the required object pronoun ("me"). Or "Mr Smith give Simon and I the belt for bad grammar" (should be "Simon and me"). And people who learnt this and in later life find some authority while avoiding dates with descriptivist grammarians might start hyper-correcting people who otherwise use their pronouns correctly (effectively? pragmatically?).
This all gets scary pretty fast, and so we might instead fudge things by using "myself" etc., which is also wrong, but less likely to draw ire: "Myself and Simon are going drinking in the park", or "the dog attacked Simon and myself". Forms like these are common in colloquial speech in Glasgow, where I live, and not a million miles from the entirely correct "I myself am going drinking", with "myself" as an emphatic pronoun.
But we can also take revenge on these ill-informed hyper-correct pedants and, to spite our expensive university educations, start dropping forms like "Me and Adam have discovered a lot of weird things since we came to Japan" into our radio conversations. It's more fun than explaining to somebody that French has disjunctive pronouns and 1066 and all that so why can't English because "Simon et moi allons au parc ce soir" is ok? It's more rootsy than opining that English speakers must have a pretty weak grip on grammatical case these days if Simon and Adam's getting in the way is enough to muck it up....
Maybe what we're really learning is that our American radio personality's gig researching weirdness in Japan is enough of an indicator of social prestige that he doesn't need the crutch of prescriptive grammar to get ahead, except of course when it came to writing the application to get the job in the first place. If they is a reasonably sophisticated user of language, perhaps with a penchant for swiping pragmatist linguists right, s/he can likely adapt xir pronouns to the situation, (un)consciously confident that good language is really all about communicative power... Or maybe instead we'll one day hear a radio programme regretfully imploring the grandchildren to invest in Grammarly after a later-life realisation that pronominal correction is a socio-political power play and hell, the history of English from the Anglo-Saxon swamp til now tells us that it's always more fun to wield power by proxy than receive hit like me, Adam and hyper-correct I did.
Such digressions aside, I'm struggling slightly to think of a situation where saying "Me and Adam" instead of "Adam and I" without a change in word order or stress changes the meaning EXCEPT in terms of social signalling -- something I gather Japanese knows about. Maybe something like "Sally saw Steven and I went home and had breakfast", versus "Sally saw Steven and me went home and had breakfast". But even here, putting a comma or pause in makes more of a difference: in "Sally saw Steven and I, went home and had breakfast", Sally gets the food, whereas in "Sally saw Steven, and me went home and had breakfast" the breakfast is likely me's comfort food.
To summarise: in informal language it's quite frequent, potentially common, potentially natural, and doesn't really affect meaning. But please don't say "Adam and me would be excellent choices for the post of Pronoun Safety Officer" in formal communication that matters.