How to POSIX-ly count the number of lines in a string variable?

The simple answer is that wc -l <<< "${string_variable}" is a ksh/bash/zsh shortcut for printf "%s\n" "${string_variable}" | wc -l.

There are actually differences in the way <<< and a pipe work: <<< creates a temporary file that is passed as input to the command, whereas | creates a pipe. In bash and pdksh/mksh (but not in ksh93 or zsh), the command on right-hand side of the pipe runs in a subshell. But these differences don't matter in this particular case.

Note that in terms of counting lines, this assumes that the variable is not empty and does not end with a newline. Not ending with a newline is the case when the variable is the result of a command substitution, so you'll get the right result in most cases, but you'll get 1 for the empty string.

There are two differences between var=$(somecommand); wc -l <<<"$var" and somecommand | wc -l: using a command substitution and a temporary variable strips away blank lines at the end, forgets whether the last line of output ended in a newline or not (it always does if the command outputs a valid nonempty text file), and overcounts by one if the output is empty. If you want to both preserve the result and count lines, you can do it by appending some known text and stripping it off at the end:

output=$(somecommand; echo .)
line_count=$(($(printf "%s\n" "$output" | wc -l) - 1))
printf "The exact output is:\n%s" "${output%.}"

Not conforming to shell built-ins, using external utilities like grep and awk with POSIX compliant options,

string_variable="one
two
three
four"

Doing with grep to match start of lines

printf '%s' "${string_variable}" | grep -c '^'
4

And with awk

printf '%s' "${string_variable}" | awk 'BEGIN { count=0 } NF { count++ } END { print count }'

Note that some of the GNU tools, especially, GNU grep does not respect POSIXLY_CORRECT=1 option to run the POSIX version of the tool. In grep the only behavior affected by setting the variable will be the difference in processing of the order of the command line flags. From the documentation (GNU grep manual), it seems that

POSIXLY_CORRECT

If set, grep behaves as POSIX requires; otherwise, grep behaves more like other GNU programs. POSIX requires that options that follow file names must be treated as file names; by default, such options are permuted to the front of the operand list and are treated as options.

See How to use POSIXLY_CORRECT in grep?