It's considered the same as an in-kind transfer for stocks from one broker to another.

Again, the understanding of "property" is evolving in this environment, and there are areas of regulatory uncertainty (such as synthetic assets from staking collateral in money markets). However, transferring from one wallet to another is pretty safe territory.

That's an implementation detail in service to controlling value on the ledger. Moving it around does not change that it represents ownership of a quantifiable amount of BTC, unless you sell it or trade it.