you can use reduce to group them by month which will give an object like

{
  12: {
    distance: 2000,
    month: 12,
    time: 4900
  },
  2: {
    distance: 1500,
    month: 2,
    time: 6400
  }
}

and using Object.values get the values array of it

let x = [{start_date: "2022-12-05T04:00:00Z" ,distance: 1000, time: 3600},{start_date: "2022-02-07T04:00:00Z" ,distance: 1500, time: 6400},{start_date: "2022-12-08T04:00:00Z" ,distance: 1000, time: 1300}]

let res = Object.values(x.reduce((acc,{start_date,distance,time})=> {
    let month = new Date(start_date).getMonth()+1
  if(!acc[month])acc[month] = {totalDistance:0,totalTime:0,month:month}
  acc[month].totalDistance+=distance
  acc[month].totalTime+=time
  return acc;
},{}))


console.log(res)

You can use a object as a dictionary and save a accumulated value of time and distance per month key. Then, reduce all keys to an array with the requested format.

const groupPerMonth = (list) => {

    const extractMonth = (stringDate) => {
        const month = new Date(stringDate).getMonth() + 1;
        return month < 10 ? `0${month}` : `${month}`;
    }
    
    const months = {};

    for (const item of list) {
        const month = extractMonth(item.start_date);

        if (!(month in months)) {
            months[month] = {
                distance: 0,
                total_time: 0,
            };
        }

        months[month].distance += item.distance;
        months[month].total_time += item.time;
    }

    const result = [];

    for (const month in months) {
        result.push({
            month,
            ...months[month]
        });
    }

    return result;
};

And test it:

console.log(
  groupPerMonth([
    { start_date: "2022-12-05T04:00:00Z", distance: 1000, time: 3600 },
    { start_date: "2022-02-07T04:00:00Z", distance: 1500, time: 6400 },
    { start_date: "2022-12-08T04:00:00Z", distance: 1000, time: 1300 },
  ])
);

Output:

[
  { month: '12', distance: 2000, total_time: 4900 },
  { month: '02', distance: 1500, total_time: 6400 }
]

There might be different solutions to this, but one way to solve this is by using the lodash library to solve this. We can first group by month, followed by mapping each grouped item and adding the distance and time values in each group using reduce:

const list = [
    {start_date: "2022-12-05T04:00:00Z" ,distance: 1000, time: 3600},
    {start_date: "2022-02-07T04:00:00Z" ,distance: 1500, time: 6400},
    {start_date: "2022-12-08T04:00:00Z" ,distance: 1000, time: 1300}
]

const grouped = _.groupBy(list, item => {
    const date = new Date(item.start_date)
    return date.getMonth() + 1
})
const groupedAndMapped = _.map(grouped, function(groupedList, date){
    return {
        month: date,
        total_distance: _.reduce(groupedList, (total, current) => {
            return total + current.distance
        }, 0),
        total_time:_.reduce(groupedList, (total, current) => {
            return total + current.time
        }, 0)
    }
})

One improvement you could do is to format the month to an "MM-YYYY" format or something similar as its possible your data set can include items with different years.