You never convert the array of strings into an array of dates:

    const dateArray = ["2000-07-13","03/24/2000", "June 7 2021", "June 13 2006"]
    const compaingDate =new Date("2000-06-13")
    let countOfDays=0
    for(let sameDate of dateArray){
    sameDate = new Date(sameDate)
        if(compaingDate.getDate()===sameDate.getDate()){
            countOfDays+=1
        }
    }
    console.log(countOfDays);

Maybe this is what you are looking for?

A shorten version:

const dateArray = ["2000-07-13", "03/24/2000", "June 7 2021"]
const compaingDate = new Date("2000-06-13")
let countOfDays = 0
dateArray.forEach(item => {
  if (new Date(item).getDate() === compaingDate.getDate())
    countOfDays++
})
console.log(countOfDays);