You are making the common mistake of assuming that [ is part of the if command's syntax. It is not; the syntax of if is simply
if command; then
... things which should happen if command's result code was 0
else
... things which should happen otherwise
fi
One of the common commands we use is [ which is an alias for the command test. It is a simple command for comparing strings, numbers, and files. It accepts a fairly narrow combination of arguments, and tends to generate confusing and misleading error messages if you don't pass it the expected arguments. (Or rather, the error messages are adequate and helpful once you get used to it, but they are easily misunderstood if you're not used.)
In your main function, the call to [ appears misplaced. You probably mean
if function "$arg"; then
...
elif ... ; then ...
By the way, for good measure, you should also always quote your strings. Use "$1" not $1, and "$arg" instead of $arg.
The historical reasons for test as a general kitchen sink of stuff the authors didn't want to make part of the syntax of if is one of the less attractive designs of the original Bourne shell. Bash and zsh offer alternatives which are less unwieldy (like the [[ double brackets in bash, which you use in your function1 definition), and of course, POSIX test is a lot more well-tempered than the original creation from Bell Labs.
As an additional clarification, your function can be simplified to just
function1 () {
! [[ "$1" =~ "^ ...some regexp... $" ]]
}
That is, perform the test with [[ and reverse its result code. (The "normal" case would be to return 0 for success, but maybe you are attempting to verify that the string doesn't match?)
for arg in "$@"with the quotes, or simplerfor arg; do ...pattern='^ ...some regexp... $'; if [[ $1 =~ $pattern ]]. Note that inside double square brackets, it's not necessary to quote variables and as glenn said, the regex (variable) should never be quoted.