There is a Scots word "shoogly" which means the same thing so I think all the Scots here would have got this immediately. We have a saying "your jaiket (jacket) is on a shoogly nail (or peg)". Meaning that your position is tenuous or threatened. Also the old trams in Glasgow used to "shoogle" or rock back and forward, and there was a joke about "Shoogle, shoogle, shoogle, all the way to Auchenshuggle" (a suburb of Glasgow, on the tram line).
On that last, Auchenshuggle itself is a name of Gaelic origin, because achadh is the root for "field" that we see in a lot of placenames beginning with "Auchen..." I don't honestly know what the "shuggle" part denotes.
Indeed, if a table is unstable because it's standing on an uneven surface, we'd say "that's really shoogly". Or indeed anything that's meant to be secure and stable, but isn't. "Wobbly" doesn't quite cut it in the same context.
I suspect the Scots word comes from the Gaelic originally. It's surprising just how much of a low background hum of Gaelic there is in Scotland, from the bilingual signs that have been adopted fairly recently, through words you just realise you've picked up somewhere and words that have migrated into Scots, to the placenames in the landscape around us. I'm constantly being surprised by words coming up that I realise I already knew. (I knew Sròn as the name of a hill, and in the context "sròn garbh", seen on a map. I didn't know it meant "nose", but that information could be added immediately without having to learn a new word. I knew "dileas" from a book I read one time, also "m'eudail" from some other book, sometimes these just stick.)
That, and being somewhat familiar with the "I am seeing" construction rather than "I see" and even "the hunger is on me" - because Gaelic speakers tend to do that when they're speaking English - does make it a bit easier I think. (Mind you, "the love is at me on you" is a new one on me!)
Yes, Wiktionary lists this meaning for press (compare Gaelic preas) in the sense of a cupboard as used in both Scotland and Ireland. It is a mystery where the word comes from as it also means 'fold' or 'bush' in Gaelic. It exists in Irish in the form prios /pɾʲɪsˠ/ but the etymology is given as 'from English'. MacBain suggests that the 'cupboard' sense of the Gaelic word comes from English, but the 'bush' sense 'probably comes from Pictish'. This is probably based on the fact that there are no words in Old Irish that start with a p unless they are obvious borrowings, such as pen and paper.
Although the Irish word is in Wiktionary and Ó Dónaill, I cannot find it elsewhere, which does make me wonder if it is a recent borrowing.
DOST, for Scots up to 1700, cites examples back to 1473 and gives the etymology as
ME and e.m.E. presse , presce (c 1400), press (c 1500), Pres n., also instrument for pressing, cupboard, printing press, F. presse (11th c. in Littré), vbl. n. f. stem of presser, L. pressāre, frequent. verb f. press-, p.p. stem of premere to press.
The Historical Thesaurus of English gives the first use in this sense as 1386.
All in all, the evidence does point to its first use being in English, although how or if this meaning evolved from the basic meaning of 'apply pressure' is not obvious.
Note, however, that the cupboard is usually used for storing clothes, folded not hanging - see linen-press (edit: and thanks to carol978657 for confirmation) and that preas also means 'fold, wrinkle, crease, rumple', so perhaps a press/preas was a cupboard for folded clothes. It could even be that the bush is a folded or rumpled tree? D
I'm sure Americans, like everyone else, have several words they could use, so yes that is another, rather less colloquial word that could be used in America or elsewhere. I say 'another' because another American, Guillermo8330, has already posted the word wobbly.
Actually, my dentist has introduced me to an even less colloquial term. She has told me one of my teeth is highly mobile.