Splitting string by the first occurrence of a delimiter

cut sounds like a suitable tool for this:

bash-4.2$ s='id;some text here with possible ; inside'

bash-4.2$ id="$( cut -d ';' -f 1 <<< "$s" )"; echo "$id"
id

bash-4.2$ string="$( cut -d ';' -f 2- <<< "$s" )"; echo "$string"
some text here with possible ; inside

But read is even more suitable:

bash-4.2$ IFS=';' read -r id string <<< "$s"

bash-4.2$ echo "$id"
id

bash-4.2$ echo "$string"
some text here with possible ; inside

With any standard sh (including bash):

sep=';'
case $s in
  (*"$sep"*)
    before=${s%%"$sep"*}
    after=${s#*"$sep"}
    ;;
  (*)
    before=$s
    after=
    ;;
esac

read based solutions would work for single character (and with some shells, single-byte) values of $sep other than space, tab or newline and only if $s doesn't contain newline characters.

cut based solutions would only work if $s doesn't contain newline characters.

sed solutions could be devised that handle all the corner cases with any value of $sep, but it's not worth going that far when there's builtin support in the shell for that.

As you have mentioned that you want to assign the values to id and string

first assign your pattern to a variable(say str)

    str='id;some text here with possible ; inside'
    id=${str%%;} 
    string=${str#;}

Now you have your values in respective variables