Are there more readable and terse method can get this list

Since it hasn't been proposed yet:

Inner[UndirectedEdge, {a, b}, {{1, 2, 3, 4, 5}, {6, 7, 8, 9, 10}}, Sequence]

{a <-> 1, b <-> 6, a <-> 2, b <-> 7, a <-> 3,
 b <-> 8, a <-> 4, b <-> 9, a <-> 5, b <-> 10}

(You did say that "I don't care (about) the ordering of its elements"! ;))

As per request:

Join@@ MapThread[Thread[#1 <-> #2] &, {{a, b}, list}] 

Flatten or Catenate to flatten list result from MapThread.

See comment by Kuba below.

How about:

{a, b} + list /. a_ + b_ :> a <-> b // Flatten