Replace a string in a string with a new line (\n) in Bash

Use ANSI C style escape sequence $'\n' to indicate newline:

$ string=xyababcdabababefab

$ echo "${string/abab/$'\n'}"
xy
cdabababefab

Or use zsh to use just \n:

% string=xyababcdabababefab

% echo "${string/abab/\n}"
xy
cdabababefab

You should use -e with echo as follows:

echo -e ${string/abab/'\n'}

From manpage:

-e     enable interpretation of backslash escapes

If -e is in effect, the following sequences are recognized:

\\     backslash

\a     alert (BEL)

\b     backspace

\c     produce no further output

\e     escape

\f     form feed

\n     new line

\r     carriage return

\t     horizontal tab

\v     vertical tab

In addition to above use newline by itself:

echo "${string/abab/
}"

note the quoting to avoid \newline substitution by space
For old bash version can be suitable:

printf "%s\n" "${string%%abab*}" "${string#*abab}"