As you can see, with moment lib, we need 2 steps to get the result: parse string to Date object, then format date object to string.

Your code - format(new Date("10-13-20"), 'MM-DD-YY') is format step, try convert a date object to a string with format template is MM-DD-YY. But your date object is not correct.

The solution is to do the same as with moment lib:

1. Parse date string to date object. Use parse

   const dateString = '10-13-20';
   const date = parse(dateString, 'MM-dd-yy', new Date()) // not MM-DD-YY
   

2. Format date object to result string. Use format

   const result = format(date, "yyyy-MM-dd'T'HH:mm:ss.SSSxxx")
   console.log(result)
   

   Result will be like (the same with moment's result in my timezone):

    2020-10-13T00:00:00.000+09:00
   

It seems that you are using Version 2.0 of date-fns, which is still in alpha (development) status currently.

What functions are available and how they work still seems to change frequently. For example, while in version v2.0.0-alpha.26 toDate() can handle string parameters, it cannot do that any longer in version v2.0.0-alpha.27. There is a new parseISO() function instead.

This should work now:

format(parseISO('2019-02-11T14:00:00'), 'MM/dd/yyyy')

However, while version 2 is still in beta, I would suggest using the stable version 1.x for now.