I have a date in this format: dd.mm.yyyy
When I instantiate a JavaScript date with it, it gives me a NaN
In c# I can specify a date format, to s
It's easy enough to split the string into an array and pass the parts directly to the Date object:
var str = "01.01.2010";
var dmy = str.split(".");
var d = new Date(dmy[2], dmy[1] - 1, dmy[0]);
You will need to create a function to extract the date parts and use them with the Date constructor.
Note that this constructor treats months as zero based numbers (0=Jan, 1=Feb, ..., 11=Dec).
For example:
function parseDate(input) {
  var parts = input.match(/(\d+)/g);
  // note parts[1]-1
  return new Date(parts[2], parts[1]-1, parts[0]);
}
parseDate('31.05.2010');
// Mon May 31 2010 00:00:00
Edit: For handling a variable format you could do something like this:
function parseDate(input, format) {
  format = format || 'yyyy-mm-dd'; // default format
  var parts = input.match(/(\d+)/g), 
      i = 0, fmt = {};
  // extract date-part indexes from the format
  format.replace(/(yyyy|dd|mm)/g, function(part) { fmt[part] = i++; });
  return new Date(parts[fmt['yyyy']], parts[fmt['mm']]-1, parts[fmt['dd']]);
}
parseDate('05.31.2010', 'mm.dd.yyyy');
parseDate('31.05.2010', 'dd.mm.yyyy');
parseDate('2010-05-31');
The above function accepts a format parameter, that should include the yyyy mm and dd placeholders, the separators are not really important, since only digits are captured by the RegExp.
You might also give a look to DateJS, a small library that makes date parsing painless...
Modified version of the accepted answer which supports value and format without 'dd' and uppercase format
function parseDate(input, format) {
    format = (format || 'yyyy-mm-dd').toLowerCase(); // default format
    var parts = input.match(/(\d+)/g), 
    i = 0, fmt = {};
    // extract date-part indexes from the format
    format.replace(/(yyyy|dd|mm)/g, function(part) { fmt[part] = i++; });
    return new Date([parts[fmt['yyyy']], parts[fmt['mm']], parts[fmt['dd']]].filter(x => x !== undefined).join('-'));
}
parseDate('10/2018', 'MM/YYYY')
t="01.01.1970"
parts = t.split(".");
for(var i = 0; i < parts.length; i++) parts[i] = parseInt(parts[i], 10);
new Date(parts[2], parts[1]-1, parts[0]);
Date defined as (Year, Month, Date)
Date()'s month parameter takes Month in Zero based index.  January = 0, february = 1, march = 2... etc  
Parsing the string to an int isn't necessary, but I dislike passing strings into functions and just hoping that JavaScript will "get it"... Sort of like how some people prefer ===
Building on CMS answer, I created this function to deal with a variable format
function parseDate(input, format) {
    format = format || 'yyyy-mm-dd'; // default format
    //Change from PHP date format to JS
    if (format == 'd/m/Y') {
      format = 'dd/mm/yyyy';
    }    
    if (format == 'd/m/Y H:i:s') {
      format = 'dd/mm/yyyy hh:ii:ss';
    }
    let date = NaN;
    if (format == 'dd/mm/yyyy') {
      let parts = input.match(/(\d+)/g),
          i = 0, fmt = {};
      // extract date-part indexes from the format
      format.replace(/(yyyy|dd|mm)/g, function(part) { fmt[part] = parts[i++]; });
      //create date for new format
      let createdDate = new Date(fmt['yyyy'], fmt['mm']-1, fmt['dd']);
      //check if dates are equal by comparing parts. The issue I had here was 
      //when I passed an invalid value for month, the output was adjustement to 
      //accomodate for the extra months
      if (
          createdDate.getFullYear() == fmt['yyyy'] && 
          createdDate.getMonth() == (fmt['mm']-1) && 
          createdDate.getDate() == fmt['dd']
      ) {
        date = createdDate;
      }
    }
    //same but taking into account hours minute and seccond
    if (format == 'dd/mm/yyyy hh:ii:ss') {
      let parts = input.match(/(\d+)/g),
          i = 0, fmt = {};
      // extract date-part indexes from the format
      format.replace(/(yyyy|dd|mm|hh|ii|ss)/g, 
      function(part) { fmt[part] = parts[i++]; });
      let createdDate = new Date(
       fmt['yyyy'], fmt['mm']-1, fmt['dd'], 
       fmt['hh'], fmt['ii'], fmt['ss']
      );
      if (
          createdDate.getFullYear() == fmt['yyyy'] && 
          createdDate.getMonth() == (fmt['mm']-1) &&
          createdDate.getDate() == fmt['dd'] && 
          createdDate.getHours() == fmt['hh'] &&
          createdDate.getMinutes() == fmt['ii'] && 
          createdDate.getSeconds() == fmt['ss']
      ) {
        date = createdDate;
      }
    }
    return date;
  }
There is no built in way to manipulate dates the way you would like.
The jQuery-UI datepicker has the functionality you want, I'm sure many other libraries have something similar.
$.datepicker.parseDate('dd.mm.yy', '31.12.2007');