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I'm used to using tag without braces like tag3 instead of tag3 in Markdown+MathJax, where it works as fine as does frac12 in normal LaTeX. But when I tried to do this in a real .tex source processed with pdflatex, I got some errors.

Here's the code:

documentclass[]article
usepackageamsmath
begindocument

[f(x)=frac32tag1]

enddocument

And this is what pdflatex says

This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Debian) (preloaded format=pdflatex)
 restricted write18 enabled.
entering extended mode
(./test.tex
LaTeX2e <2017-04-15>
Babel <3.18> and hyphenation patterns for 3 language(s) loaded.
(/usr/share/texlive/texmf-dist/tex/latex/base/article.cls
Document Class: article 2014/09/29 v1.4h Standard LaTeX document class
(/usr/share/texlive/texmf-dist/tex/latex/base/size10.clo))
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty
For additional information on amsmath, use the `?' option.
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty))
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty)
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty))
No file test.aux.
Runaway argument?
1] 
! Paragraph ended before tag was complete.
<to be read again> 
 par 
l.6 

When I surround the 1 with braces, it compiles fine, and I get the expected PDF output.
So I wonder, why does tag require braces while frac doesn't? Is there some difference in "type" of the object, like e.g. tag being a package-supplied facility while frac is from the language core (doesn't need usepackage at least) or something like that?

The macro frac is defined (robustly) as

frac#1#2 -> begingroup #1 endgroup @@over #2

where @@over is (a copy of) the TeX primitive. So in this case you can indeed write frac12 and it will be the same as frac12. A lot of people, included myself, will tell you not to do this. (Although for such short fractions I don't always follow my own advice...)

On the other hand, at the beginning of displaymath tag is let to tag@in@display, whose definition reads

deftag@in@display#1#relaxtag@in@display@a#1

The tricky bit is the # before the opening brace of the replacement text. TeX looks for the first argument up to an opening brace (later it will check if this first argument is a star *). For details on how this mechanism works I refer to the questions Grab to #{ macro arguments (in particular Martin Scharrer's answer) and Macros with # as the last parameter. But the short answer is: TeX is scanning for an opening brace to collect the argument #1 and at some point finds the empty line (which is converted to a new paragraph), which isn't allowed, and hence the Paragraph ended error.