grep com expressões regulares estendidas:
grep -iE '^[^[:blank:]]+[[:blank:]]+([^aeiou]*[aeiou]){2}[^aeiou]*\>' file
grep com pcre
grep -iP '^\S+\s+([^aeiou]*[aeiou]){2}[^aeiou]*\b' file
perl (sinceramente, fiz isso independentemente do comentário da steeldriver)
perl -ane 'print if (lc($F[1]) =~ tr/aeiou/aeiou/) == 2' file
awk
awk '{col2 = tolower($2); gsub(/[aeiou]/,"",col2)} length($2) - length(col2) == 2' file