The first tomatoes to reach Italy in the 16th century happened to be of a yellow variety, which is why they got called “pomo d’oro.” This has also been borrowed into Arabic as bandurah.
It’s worth noting (OP touches on it) that potatoes themselves are named for something else - the sweet potato.
Sweet potatoes were called ‘batata’ (and the modern botanical name reflects this - Ipomoea batatas) - when earth-apples came along, they were named potatoes because of their resemblance to sweet potatoes.
Yeah I’d say so. The only reason I know it, or any Polish, is the Polish supermarket across the road from my store is cheap. Sometimes though it’s a crapshoot what I’m actually buying. Fasola = beans too. Yay, I’m learning 
So it’s do re mi beans ti do?
Well, they are the musical fruit.
That’s pretty much the way I start learning any other language…through their food. I probably know about a hundred words in Hindi, for instance, and the bulk of them are food related. Even though I grew up with Polish, there’s a lot of words for certain ingredients that I know in, say, Hungarian (where I lived for a few years) that I don’t know in Polish.
Because it’s of Andean highland origin and basically a cool climate crop. It grows well at high elevations and in the temperate zone, but not in tropical lowlands. So it wasn’t able to diffuse from South America to Middle America across the gap in the mountain chain in Panama.
The tomato and most other New World crops are of lowland origin and were able to spread more widely.
^ This.
Ah, if only kielbasa were kolbasz! Unbelievably, there are no more Hungarian butchers left in NYC. 
Somebody once wrote:
I give, a king, to me, she does, alone, up there, yes see, I double give…I give to me alone I trouble give!
Somebody else once wrote a mass with the notes la, sol, fa, mi, re, and performed it addressed to an Italian King asking if he could take a vacation.
<exits time machine> … OK, if I did this right, everyone the world over is now familiar with the one and only names for:
tree apple
ground apple
orange ground apple
yellow tree apple
tree orange
vine apple
orange vine apple
squashed tree apple
vine squashed apple
We don’t know they didn’t. My table in the OP wasn’t exclusive, I only put in the native words that appeared to lead to European words. That said, it’s a cool climate crop and probably wouldn’t survive in the hot dry limestone of central Mexico.
Indeed I’m not aware of any evidence the Inca and the Aztecs directly communicated or traded. The panama isthmus was as good a barrier to travel as swamps or steep mountains, I believe.
According to wikipedia ‘potato’ is a mix of the two origin words I put in, ‘batata’ and ‘papa’:
So the vowel/consonant pattern came from one word and the plosive ‘p’ from the other.
There was quite some confusion between sweet potatoes and potatoes:
But it seems there’s no particular reason for the exceptions noted in the OP?
How do ya like them apples?
Is there any human language where potatoes are “earth bread”, like they are for the dwarves?
They do seem more like a bread than a fruit.
Throw “yams” into the mix and it’s serious Babel.
This lends insight into the curse by Shoshone chief Washakie in 1887 when Indian commissioners were trying to make the western hunter-gatherer tribes start farming under the Dawes act. Chief Washakie told them: “God damn a potato!” Even though it’s native to the Americas, it was still foreign to North American tribes.
North America and South America were not joined together before Panama was raised up to connect them. Panama is not of the continental crust but uplifted oceanic crust. Which has to do with why the land bridge between the two continents is tropical lowlands in the Darién Gap, where the Cordillera system chain is interrupted, and the potato did not indigenously transfer to North America.
In case anyone wonders what this is talking about, it’s the do-re-mi syllables which coincidentally form words in Italian.
do = I give
re = king
mi = to me
fa = he/she/it does
sol (short for sola) = alone
la = there
si = yes
The scale.
do = I give
mi = to me
sol (short for sola/o) = alone
The tonic major triad.
‘There alone make me, king’
How did we get on this starting from potatoes?
Yes.
The first scale/sentence and chord/sentence are from Finnegans Wake. A character is imagining what joy it would be to smoke a pipe all alone surrounded by music. “I’d like own company best … Isn’t that lovely though? I give to me alone I trouble give!” Closer to Joyce’s intent is that character sees music “twined me abower”: with the packed-in meaning that he is like a tree surrounded by ivy/wrapt/wrappedbetween music.
This is a way to see the pleasure in reading the Wake, and one key to the extraordinary expressivity of its incessant “simple” punning.
The “Great C-Major Chord of Life”–a bit of a poem by someone–as noted (heh) by Joanna and rarely noticed by commentators, appropriately caps the passage, and is in miniature another instance of the Wake’s unshakable structural theme of the unceasing joy of the world.
The full passage is here:
I’d like own company best, with the help of a norange and bear, to be reclined by the lasher on my logansome, my g. b. d. in my f. a. c. e., solfanelly in my shellyholders and lov‘d latakia the benuvolent, for my nosethrills with the jealosomines wilting away to their heart’s deelight. and the king of saptimber letting down his humely odours for my consternation, dapping my griffen, burning water in the spearlight, or catching trophies of the king‘s royal college of sturgeons by the armful for to bake pike and pie while,O twined me abower in L’Alouette’s Tower, all Adelaide’s naughtingerls, juckjucking benighth me, I’d tonic my twittynice Dorian blackbudds off my singasongapicoolo to pipe musicall airs on numberous fairyaciodes. I give, a king, to me, she does alone up there, yes see, I double give till the spinney all eclosed asong with them. Isn’t that lovely though? I give to me alone I trouble give!
The second part of Joanne’s response is also correct: it is the usual title of a musical setting of the mass Missa La so fa re mi, by Josquin de Prez. (Josquin’s fame in the 15th century was commensurate with Joyces now.) The syllables have also been glossed “Lascia fare me” (“Let me go” in Italian).
From this path. Ain’t thread drift great?
Apple-related nomenclature is not restricted to French and Italian, as the OP suggests. To be more exact, there’s a number of different languages with some variant of ‘earth-apple’ for potato, and there’s also a number of different languages that have apple-based words for **orange **(the fruit, not the color): sinaasappel (Dutch; from ‘Chinese Appel’); Apfelsine (German; similar etymology); pomeranč (Czech, ‘orange appel’, from the French). It does seem to be the case that words for **tomato **are either going to be some form of tomato, or some directly copied form of the italian ‘appel of gold’ (eg in the Russian and in the Polish). The Czech language is an exception here - my Czech dictionary does turn up ‘rajské jablíčko’ (paradisical little apple) but I’ve never heard that; rajče is more common. Still, there appears to be some sort of apple-root there that is not ‘apple of gold’.
Some more data points:
Dutch Aardappel (Earth Apple); Tomaat
As has been pointed out, in some cases, Germans also use Erdapfel instead of Kartoffel (wiki).
In some Slavic languages:
Czech Brambora; Rajče
Russian Kartofel’; Pomidor
Czech brambora actually refers to the region of Brandenburg, in East Germany, from whence they apparently procured their potatoes. No link to apples there, though. I do seem to recall that in some Slavic language there are words for potato like ‘zemlyak’ or ‘zemak’, related to ‘zeme’, earth - but maybe I’m just making that up.
Wait, “Tomaat” –> “Tomato”? Back in the crossing paths, but in one language.