Instead of concatenation:

str=""
str+="This is a\n"
str+="multiline string"

You can use a heredoc:

cat <<EOF
This is a
multiline string
EOF

However, heredocs are meant for standard input. To store in a variable, you need a command that reads from standard input and assigns it.

That's what read does:

read -r -d '' str <<EOF
This is a
multiline string
EOF

The -r flag prevents backslash escapes from being interpreted, and the -d '' flag means read until EOF (empty delimiter).