Создайте массив месяцев с вложенными днями между диапазоном дат

const
  monthFrmt = new Intl.DateTimeFormat('en', { month: 'long' }),
  startDate = new Date('2023-05-01T00:00:00'), // May 1st
  endDate = new Date('2023-06-03T00:00:00'),   // June 3rd (inclusive)
  dateFormatterFn = (date) => `${ordinal(date.getDate())} ${monthFrmt.format(date)}`;

const main = () => {
  const dates = generateDates(startDate, endDate, dateFormatterFn);
  console.info(dates);
};

const generateDates = (startDate, endDate, dateFormatterFn) => {
  const results = [], currDate = new Date(startDate.getTime());
  while (currDate <= endDate) {
    const key = monthFrmt.format(currDate);
    if (results[results.length - 1]?.key !== key) results.push({ key, values: [] });
    results[results.length - 1].values.push(dateFormatterFn(currDate));
    currDate.setDate(currDate.getDate() + 1);
  }
  return results.map(Object.values);
}

const rules = new Intl.PluralRules('en', { type: 'ordinal' });
const suffixes = { one: 'st', two: 'nd', few: 'rd', other: 'th' };
const ordinal = (number) => `${number}${suffixes[rules.select(number)]}`;

main();

.as-console-wrapper { top: 0; max-height: 100% !important; }

let start = new Date("05/01/2023");
let end = new Date("06/15/2023");
let arr = [];
let months = [];
const dayFormatter = new Intl.DateTimeFormat('en-US', {day: "numeric"});
const monthFormatter = new Intl.DateTimeFormat('en-US', {month: 'long'});

while (start <= end) { //get an array of date objects, one per day
  arr.push(new Date(start));
  start.setDate(start.getDate() + 1);
};

arr.forEach(function(date){ //get array of months
  let m = monthFormatter.formatToParts(date).find((o) => o.type == "month").value;
  if (!months.includes(m)){
    months.push(m);
  }
})

months.forEach(function(val,index){ //nest an array in each month
  months[index]=[val,[]];
});

months.forEach(function(month){
  arr.forEach(function(date){ //fill the nested arrays
    let m = monthFormatter.formatToParts(date).find((o) => o.type == "month").value;
    if (m === month[0]){
      let d = dayFormatter.formatToParts(date).find((o) => o.type == "day").value;
      month[1].push(`${ordinal(d)} ${m}`); //IF YOU WANT 1st May format
      //month[1].push(date); //IF YOU WANT THE DATE OBJECT INSTEAD
    }
  });
})
console.info(months)

function ordinal(number) {
  const pr = new Intl.PluralRules("en-US", { type: "ordinal" });
  const ordinals = {one:'st',two:'nd',few:'rd',many:'th',zero:'th',other:'th'}
  return `${number}${ordinals[pr.select(number)]}`;
}