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
   

I use the esm version of date-fns and you can use the same type of formats that moment uses :

import { format } from 'date-fns/esm'
import { enUS, fr } from 'date-fns/esm/locale'

I'll store the locales in an object :

this.dateLocales = { fr: fr, en: enUS }

and use these formats :

LT: 'h:mm aa',
LTS: 'h:mm:ss aa',
L: 'MM/DD/YYYY',
LL: 'MMMM D YYYY',
LLL: 'MMMM D YYYY h:mm aa',
LLLL: 'dddd, MMMM D YYYY h:mm aa'

So you can do :

format(
  new Date(2014, 6, 2),
  'LL',
  {locale: this.dateLocales.fr}
)

Those formats are localised

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.