Signing into your Xbox Live (Microsoft) account on a different Xbox console is fully supported and seamless. Here's what happens:
Xbox uses Cloud Saves (automatic, unlimited, free). Launch the game on the new console → "Syncing data" → Continue where you left off. Works across Xbox One, Series X|S, and PC (Play Anywhere titles).
Result: Your points, Gold, Gamertag, and library follow you instantly. Just re-download the games you want to play.