ENS App and Explorer are on the Sepolia testnet. Features may change before mainnet.
The owner of an ENS name is the wallet that holds its token. Every registered name is an NFT: a token that one wallet holds at a time. Hold the NFT and you own the name; transfer it and the name goes with it. Ownership is recorded on Ethereum, not in any app.
Owning a name and being allowed to act on it are different things: the owner can hand out roles without giving up the name.
Owner, Admin and User
Control of a name has three levels:
Owner: the wallet that holds the NFT. It has full authority and can grant or revoke roles for other wallets.
Admin: a permission on one action, such as changing its resolver (the contract that stores its records) or managing its subnames. An Admin can do that action and can also grant it to other wallets.
User: a permission on one action. A User can do that action but cannot pass it on. Admin is the higher of the two.
When you register a name, all three line up: your wallet owns the name and holds Admin and User on every action. They differ only once you grant a role to another wallet. For the roles and how to grant them, see Who can manage my name? Roles in ENS Explorer.
Check what role you have
Open the name in ENS Explorer and select the Roles tab. It lists every wallet with a permission on the name, with a tick under Admin or User for each role. That tab is the truth about who can do what today.
In the ENS App, a name where you hold a role but not the NFT sits under the dashboard's MANAGED filter. A banner there points you to ENS Explorer for roles.
Common questions
Can someone else hold a role on my name?
Yes. That is what roles are for. As the owner you can grant User or Admin on specific actions to other wallets while you keep the NFT. Typical case: a helper who manages subnames on a community name for its members. Letting a wallet edit records is a separate list, on the resolver: see Who can edit my records? Resolver roles in ENS Explorer. Grant and revoke roles from Add user on the Roles tab in ENS Explorer. The NFT, and so the final say, stays with you unless you transfer it.
What kind of token is the name?
A plain .eth name from the older ENSv1 system is an ERC-721 token; every ENSv2 name is an ERC-1155 token. Those are just two token standards. It makes no difference to ownership: whoever holds the token owns the name.