I would make a new object with an old_position key, based on this value you can decide what color you want to use.

const oldLeaderboard = [
  { position: '0', id: '5f862368df7d2700174c9df7' },
  { position: '1', id: '6111697148108400154a7cf9' },
  { position: '2', id: '610c1a8b132a880015c16a9d' },
  { position: '3', id: '5f8625e7df7d2700174c9df8' },
  { position: '5', id: '61153dadd7db540015836e8e' },
  { position: '4', id: '5fae7f6526ec3b0017710e60' },
  { position: '6', id: '5f849db0b0793b00177e2317' },
  { position: '7', id: '5fa85aba9c4a1900177e5cf9' },
  { position: '8', id: '6144d97107efcf001584706e' },
  { position: '9', id: '5e4463a395c832405e7effc5' }
]

const newLeaderboard = [
  { position: '0', id: '5f862368df7d2700174c9df7' },
  { position: '1', id: '610c1a8b132a880015c16a9d' },
  { position: '2', id: '5fd7b30abb015f0017eda459' },
  { position: '3', id: '5fce234f9a61c100175f4b6d' },
  { position: '4', id: '5fca23930aa5e9001797a885' },
  { position: '5', id: '5f806a8b045fdf0017734fe8' },
  { position: '6', id: '5fc10c379378110017477dca' },
  { position: '7', id: '61153dadd7db540015836e8e' },
  { position: '8', id: '5f9b2c579050d2001785a253' },
  { position: '9', id: '5e4463a395c832405e7effc5' }
]

const leaderBoard = newLeaderboard.map( item => ({
  posisiton: item.position,
  old_position: oldLeaderboard.filter(old => old.id === item.id)[0]?.position || null,
  id: item.id
}))

console.log(leaderBoard)

change the data-structure use HashMap (JSON) instead of array with id as key and position as value.

By doing you would reduce the total comparisons for an element and then simply compare the positions.

const newLoop =[
  { position: '0', id: '5f862368df7d2700174c9df7' },
  { position: '1', id: '610c1a8b132a880015c16a9d' },
  { position: '2', id: '5fd7b30abb015f0017eda459' },
  { position: '3', id: '5fce234f9a61c100175f4b6d' },
  { position: '4', id: '5fca23930aa5e9001797a885' },
  { position: '5', id: '5f806a8b045fdf0017734fe8' },
  { position: '6', id: '5fc10c379378110017477dca' },
  { position: '7', id: '61153dadd7db540015836e8e' },
  { position: '8', id: '5f9b2c579050d2001785a253' },
  { position: '9', id: '5e4463a395c832405e7effc5' }
]
const newLoopJSON={}
for (let i of newLoop){
  newLoopJSON[i.id]=i.position
}
const lastLoop= [
  { position: '0', id: '5f862368df7d2700174c9df7' },
  { position: '1', id: '6111697148108400154a7cf9' },
  { position: '2', id: '610c1a8b132a880015c16a9d' },
  { position: '3', id: '5f8625e7df7d2700174c9df8' },
  { position: '4', id: '5fae7f6526ec3b0017710e60' },
  { position: '5', id: '61153dadd7db540015836e8e' },
  { position: '6', id: '5f849db0b0793b00177e2317' },
  { position: '7', id: '5fa85aba9c4a1900177e5cf9' },
  { position: '8', id: '6144d97107efcf001584706e' },
  { position: '9', id: '5e4463a395c832405e7effc5' }
]

for(let {id,position} of lastLoop){
    if(newLoopJSON[id]){
        let new_POS=newLoopJSON[id]
        let difference=new_POS.id-id
        if(difference==0){ 
            // no change in new position and old position
            console.log("black")
        }
        else if(difference<0){
            // position increased
            console.log("green")
        }else{
            // position decreased
            console.log("red")
        }
        
    }
}

// ...
const colorRules = [
  {
    color: "green",
    value: (oldP, newP) => +newP > +oldP
  },
  {
    color: "black",
    value: (oldP, newP) => +oldP === + newP
  },
  {
    color: "red",
    value: (oldP, newP) => +oldP > + newP
  },
]

const result = newLoop.map(player => {
  const oldHistory = lastLoop.filter(item => item.id === player.id);

  if (oldHistory.length) {
    const colorRulesMatch = colorRules.map((c) => ({
      ...c,
      value: c.value(oldHistory[0].position, player.position)
    }));

    const { color } = colorRulesMatch.filter(c => c.value)[0];

    return {
      ...player,
      color
    }
  } else {
    return {
      ...player,
      color: "black" // If player wasn't found in last loop
    }
  }
});

console.log(result);
// [
//   { position: '0', id: '5f862368df7d2700174c9df7', color: 'black' },
//   { position: '1', id: '610c1a8b132a880015c16a9d', color: 'red' },
//   { position: '2', id: '5fd7b30abb015f0017eda459', color: 'black' },
//   { position: '3', id: '5fce234f9a61c100175f4b6d', color: 'black' },
//   { position: '4', id: '5fca23930aa5e9001797a885', color: 'black' },
//   { position: '5', id: '5f806a8b045fdf0017734fe8', color: 'black' },
//   { position: '6', id: '5fc10c379378110017477dca', color: 'black' },
//   { position: '7', id: '61153dadd7db540015836e8e', color: 'green' },
//   { position: '8', id: '5f9b2c579050d2001785a253', color: 'black' },
//   { position: '9', id: '5e4463a395c832405e7effc5', color: 'black' }
// ]