What if a person has your private key?
Then they have independent control over all the unspent Bitcoins associated with that private key. Once they have the private key there is nothing you can do to rescind their control (other than moving the Bitcoin into a new wallet before the other person does)
How can that person interact with your wallet?
They don't need to.
The only valuable thing in your wallet are the private keys.
Bitcoins are "stored" as unspent transaction outputs (UTXOs) in the blockchain - but the blockchain is completely public. Every full node (wallet, miner etc) has a copy of the blockchain.
Does he have to know which wallet you use as well ...
No, the other person doesn't need to know anything else.
... or can he use just the private key to 'log in'?
They don't use the private key to log in.
They can import the private key into any wallet software and use the private key to prove to all other bitcoin nodes that they have the right to spend any unspent bitcoin associated with that private key.
Knowing the private key allows them to construct a transaction spending your Bitcoins, they then broadcast that transaction to miners who add the transaction to the blockchain thus spending the Bitcoins (i.e. moving Bitcoins into another wallet which has a different private key)