I've tried to define custom mode for highlight custom markdown syntax with easyMDE like this:

CodeMirror.defineMode("mymode", function () {
  return {
    token: function (stream, state) {
      if (stream.match("aaa")) {
        return "style1";
      } else if (stream.match("bbb")) {
        return "style2";
      } else {
        stream.next();
        return null;
      }
    },
  };
});

window.easyMDE = new EasyMDE({
  element: $('#textarea')[0],
  parsingConfig: {
    allowAtxHeaderWithoutSpace: true,
  },
  overlayMode: {
    mode: CodeMirror.getMode({}, 'mymode'),
    combine: true,
  },
});

jsFiddle with example https://jsfiddle.net/6ycwmL04/3/