Functions in a different context are replaced with infix forms of their namesake in System`

It seems like the StandardForm (default for output) of Times in any context (or Plus, etc) is with the * and + symbols.

But your assumption isn't wrong I think, the symbols * and + are mapped to System Times and System Plus. Try

PrependTo[$ContextPath, "blo`"];
blo`Times[a_, b_] := 8;

FullForm@MakeExpression[RowBox[{"a", "*", "b"}], StandardForm]

I think that this goes against Mathematica policy that "StandardForm generates output that gives a unique and unambiguous representation of Mathematica expressions, suitable for use as input"

The problem seems to be with ToBoxes:

Outer[#1[#2[a,b]]&,{ToString[#,OutputForm]&,ToBoxes},{Times,foo`Times}]
(*
==> {{"a b", "foo`Times[a, b]"}, {\(a\ b\), \(a * b\)}}
*)

Note that Times and foo`Times are still different in that for Times, no * is generated.

When typing to the kernel directly, OutputForm is used, and foo`Times[a,b] gets printed:

In[1]:= {Times[a,b], foo`Times[a,b]}

Out[1]= {a b, foo`Times[a, b]}