A wallet stores keys, not coins. The balance lives on the blockchain; the wallet is the tool that proves you control it. That is why the same holdings reappear in a new wallet the moment you restore the keys - nothing moved.
The important distinction is who holds those keys. If a platform holds them for you, you have an account and a counterparty. If you hold them yourself, you have full control and full responsibility, including for the mistakes. Both are legitimate. Confusing one for the other is what causes trouble.
Binance Academy covers this in more depth: https://www.binance.com/en/academy/glossary/wallet
Trading crypto from Dubai since 2019.
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