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

As you are using the FP version to format a date you need to use the formatWithOptions function to pass the locale options.

import format from "date-fns/fp/formatWithOptions";

It has a slightly different function signature then the format function, where you need to pass the options as the first argument.

const formatMonth = format({ locale: de }, "MMM");

The P patterns to format deal with localization: https://date-fns.org/v2.25.0/docs/format

Looks like you would want the PP variant and can be used like this:

import { caLocale } from 'date-fns/locale/ca'

const result = format(new Date(2021, 0, 12), "PP", {
  locale: caLocale
})

This has been released in version 2.29: https://date-fns.org/v2.29.0/docs/setDefaultOptions

// Set global locale:
var setDefaultOptions = require('date-fns/setDefaultOptions')
import { es } from 'date-fns/locale'
setDefaultOptions({ locale: es })

Although date-fns does not support a native method to do partial overwrites (yet), you can do the following, to do some manual tweaking (shown here by the german locale):

import { formatRelative } from 'date-fns';
import { de } from 'date-fns/esm/locale';

const formatRelativeLocale = {
  lastWeek: '[letzten] dddd [um] LT',
  yesterday: '[gestern um] LT',
  today: '[heute um] LT',
  tomorrow: '[morgen um] LT',
  nextWeek: 'dddd [um] LT',
  other: 'L LT', // Difference: Add time to the date
};

const locale = {
  ...de,
  formatRelative: token => formatRelativeLocale[token],
};

const text = formatRelative(date, new Date(), { locale });