Hfrhyu

Continuing https://math.meta.stackexchange.com/a/22167/, as far as I understand, all the four of ℕ, mathbbN, BbbN, symbbN work now, and BbbN is advised against. Is there any reasonably default context (e.g., a self-constructed context that would redefine these macros and symbols wouldn't count) in which some of ℕ, mathbbN, BbbN, symbbN produce different results than some others when using amssymb+unicode-math+xelatex? Compiling the example

documentclassbook
usepackagefontspec
usepackageamssymb
usepackageunicode-math
usepackagemicrotype
setmainfontTeX Gyre Termes
setsansfontTeX Gyre Heros[Scale=0.88]
setmonofontTeX Gyre Cursor
setmathfontTeX Gyre Termes Math
setmathfontAsana Math[
 range=setminus,
]
setmathfontXITSMath-Regular[
 Extension=.otf,
 range="2A3E,
 BoldFont=XITSMath-Bold,
]
begindocument
(ℕ mathbbN BbbN symbbN)
enddocument

with xelatex, e.g., I get visibly indistinguishable letters

ℕℕℕℕ

I cannot distinguish them either when I put them as subscripts or superscripts.

Moreover, is there a consensus in the xe[La]TeX world to name any of these ways as the standard way to denote the set of natural numbers?

(Of course, I leave aside the question whether the zero should belong to this set or not; it could flame up a war here and is up to the author anyway.)

The difference is mainly historical. BbbN was created for the original amsfonts, pre-LaTeX; it should be considered obsolete now.
(Oops! @egreg points out in a comment that BbbN has been defined for unicode-math, so I was thinking of BbbN. That surely should be considered obsolete.)
The original LaTeX equivalent is mathbbN, and should still be reliable.

symbbN was defined for fonts developed after the blackboard bold alphabet was added to Unicode.

The symbol itself (which I can't represent because it's not available on the aged laptop I'm using) depends on having a utf-8 capable input device, and is not available for pdflatex, which is still limited to 8-bit input.

All forms are equivalent, and the one you use depends on which flavor of LaTeX you're using. There may also be some restrictions associated with the publisher, if you're submitting your document for publication.

This may not give an unambiguous answer to your question, but it should give you some idea of how the development of the blackboard bold fonts and their support affects the decision of which should be used in what circumstances.

If you modify your file to have

showoutput
(ℕ mathbbN BbbN symbbN showlists)

Then you get

mathord
.fam0 ℕ
mathord
.fam0 ℕ
mathord
.fam0 ℕ
mathord
.fam0 ℕ
### horizontal mode entered at line 20

Four identical N, same font and same math class (mathord).

I would say use ℕ if you like Unicode input and symbbN if you prefer ASCII TeX command markup. So they are the preferred forms, but as they are all the same thing it doesn't matter much which you use.

Of course other font setups may make things differ. In general symxx will give you characters from the same font using the math alphabet ranges, whereas mathxx might do that or might (as in classic tex) use a different font.

tl;dr

It's completely the same.

Why do those four inputs produce the same output?

In unicode-math-table.tex we find

UnicodeMathSymbol"02115BbbNmathalpha/bbb n, open face n

Every Unicode code point relevant for math has a name, so that unicode-math can do, in this case, the equivalent of

Umathchardef`ℕ = "7 "0 "02115

(the second number could change in case range=bb is used to select a different font for these characters).

If you add showmathbb to your sample TeX file (after begindocument), you'll get

> mathbb=long macro:
->symbb .

This almost answers your question. At least we know that

1. typing ℕ or BbbN is the same


2. typing mathbbN or symbbN is the same

It only remains to discover what's the relationship between the two cases above. Simple: symbbN does BbbN. Not really by chaining N to Bbb, but something like that (it's more complicated because one can use range=bb to use a different font for blackboard bold letters).

Now we know that typing

$ℕ BbbN mathbbN symbbN$

is exactly the same. The alias name mathbb for symbb is for backwards compatibility with older code.

Some explanation is in order. unicode-math used to have just mathXX commands. However, it was realized that distinguishing between mathXX and symXX is necessary. The first form is about words used in math, the second form for single characters (and doesn't enforce ligatures if used for more characters in a row); these forms can point to different fonts. Typically, for instance, mathbf will use the boldface text font, whereas symbfx will use mbfx, pointing to U+1D431 in the math font.

While the distinction is necessary for boldface, in the case of blackboard bold there is no usage of it as a text font, so no distinction is made between mathbb and symbb, by default. You (or a package) might redefine mathbb to do something else (not that I recommend it).

What's the preferred form?

I'd avoid BbbN and probably prefer symbb for newer documents, unless it's possible to directly type in ℕ.