Otdfbt

What is the purpose of $:$?

We know that in math mode one should write in natural language and not being so strict with math notation.

For example: $f:Ato B$ is not "correct", the correct way would be $fcolon Ato B$ (and variants).

I do not remember any math book that uses the colon as part of a sentence in math mode.

The only 2 uses I can give to $:$ are when learning about divisions ($2:4=1:2$) and scales ($1:2$).

Does (La)TeX give a meaning of : in math mode?

The : is a relational symbol in TeX. In the TeXbook it is used in conjunction with =.

Plain TeX treats the four characters |=|, |<|, |>|, and |:| as
``^relations'' because they express a relationship between two
quantities. For example, `$x<y$' means that $x$~is less than~$y$.
Such relationships have a rather different meaning from binary
operations like $+$, and the symbols are typeset somewhat differently:
beginmathdemo
|$x=y>z$|&x=y>zcr
|$x:=y$|&x:=ycr
|$xle yne z$|&xle yne zcr
|$xsim ysimeq z$|&xsim ysimeq zcr
|$xequiv ynotequiv z$|&xequiv ynotequiv zcr
|$xsubset ysubseteq z$|&xsubset ysubseteq zcr
endmathdemo

In the comments, egreg pointed out another example in the TeXbook, where : is used to separate the condition from the set variable.

beginmathdemo
|$,xmid x>5,$|&,xmid x>5,cr
|$,x:x>5,$|&,x:x>5,cr
endmathdemo
(Some authors prefer to use a ^colon instead of `$mid$', as in the second
example here.)

The colon has the exact same meaning in Plain TeX and LaTeX, as can be verified using this simple document:

showthemathcode`:
csname @@endendcsname
bye

The output is

$ pdftex test.tex
This is pdfTeX, Version 3.14159265-2.6-1.40.20 (TeX Live 2019) (preloaded format=pdftex)
 restricted write18 enabled.
entering extended mode
(./test.tex
> 12346.
l.1 showthemathcode`:

? 
 )
No pages of output.
Transcript written on test.log.
$ pdflatex test.tex
This is pdfTeX, Version 3.14159265-2.6-1.40.20 (TeX Live 2019) (preloaded format=pdflatex)
 restricted write18 enabled.
entering extended mode
(./test.tex
LaTeX2e <2018-12-01>
> 12346.
l.1 showthemathcode`:

? 
 )
No pages of output.
Transcript written on test.log.

Here the same table rendered in LaTeX:

documentclassarticle
usepackagearray
usepackageamsmath
usepackagelistings
lstsetbasicstyle=ttfamily
begindocument
lstMakeShortInline|
begintabularl>$l<$
|$x=y>z$| & x=y>z \
|$x:=y$| & x:=y \
|$xle yne z$| & xle yne z \
|$xsim ysimeq z$| & xsim ysimeq z \
|$xequiv ynotequiv z$| & xequiv ynotequiv z \
|$xsubset ysubseteq z$| & xsubset ysubseteq z \
endtabular
enddocument

Quite often, one sees : employed in place of a vertical bar (|) to denote conditioning on some event or condition. E.g., both $E(X : X>0)$ and $E(Xmid X>0)$ would express the "expectation of the random variable X conditional on this random variable taking on positive values".

By default, TeX treats both : and mid -- but not | and vert -- as "relational" operators and inserts an amount of whitespace around them that's also used for other relational operators such as <, =, and >.

For other, i.e., non-conditioning math uses of the colon symbol, one should employ colon, coloneq (or, depending on the package that's in use, coloneqq, etc.