const regex = /\*\*\[(.+)\]> \[([^*]+)\]/gm;
const str = `**[Никнейм]> [сообщение]**
**[Никнейм2]> [сообщение2]**`;
let m;
while ((m = regex.exec(str)) !== null) {
if (m.index === regex.lastIndex) {
regex.lastIndex++;
}
if(m.length < 3) continue;
const nickname = m[1];
const message = m[2];
console.log(nickname);
console.log(message);
}