mjhjmtu

beginalign*
sum_j=1^m_0-1,(-1)^j-1hspace-3ex 
sum_substack k_m_0 +1 leq k_m_0 leq cdots leq k_j+2 leq k_1 leq boldsymbolell \ k_2 < cdots < k_j < k_j+1 leq k_1 
= sum_ k_m_0 +1 leq k_m_0 leq cdots leq k_2 leq k_1 leq 
 boldsymbolell, tag(3) labeleq:3
endalign*
which is very close to our goal, equation eqrefeq:2. 
Comparing equation eqrefeq:3 with equation eqrefeq:2, we discover that both have the identical 1tsst through $(m_0 -2)$tsth terms ($1 leq j leq m_0-2$). 
The only difference of the two is that the $(m_0 -1)$tsth term in equation eqrefeq:3 is ...
enddocument

I try to put a tag next to the equation, but the result comes out as:

The tag is not displayed correctly. Also, the eqrefeq:3 are displayed in the same, wrong way. (I omit the code having eq:2 and eq:1.)

tag<tag> takes a mandatory argument. If you don't supply the surrounding braces, then it takes the next token as the argument.

In your case, tag(3) doesn't use the suggested format tag<tag>, so it turns into tag(3), numbering the equation with ( and leaving 3) in the input stream. You're looking for tag3:

documentclassarticle

usepackageamsmath,amssymb

begindocument

beginalign*
 sum_j = 1^m_0 - 1,(-1)^j - 1hspace-3ex 
 sum_
 substack
 k_m_0 + 1 leq k_m_0 leq cdots leq k_j + 2 leq k_1 leq boldsymbolell \
 k_2 < cdots < k_j < k_j+1 leq k_1 
 
 
 = sum_ k_m_0 + 1 leq k_m_0 leq cdots leq k_2 leq k_1 leq boldsymbolell , tag3 labeleq:3
endalign*
which is very close to our goal, equation~(2). Comparing equation~eqrefeq:3 with equation~(2), we discover that both have the identical 
1textsuperscriptst through $(m_0 - 2)$textsuperscriptth terms ($1 leq j leq m_0 - 2$). The only difference of the two is that the 
$(m_0 - 1)$textsuperscriptth term in equation eqrefeq:3 is ldots

enddocument