You can do this with a combination of sortand join. Direct approach
join -j2 <(sort -k2 file1) <(sort -k2 file2)
but it looks a little different than what you are looking for. It just shows the common join field, and then the rest of the fields from each file
"1431_at" "3973" 2.52832098784342 "653" 2.14595534191867
"207201_s_at" "1826" 2.41685345240968 "1109" 2.13777517447307
, , join
join -o 1.1,1.2,1.3,2.1,2.2,2.3 -j2 <(sort -k2 file1) <(sort -k2 file2)
-o FILENUM.FIELDNUM.
, <(), , POSIX sh, , POSIX sh.