Trying to repair a leaking faucet handle, but can't determine the brand
That looks a moderately standard tap

What I'd do,


Shut off water to the faucet (isolation valve below or elsewhere).
Use pliers to pull the spring-clip off.
Use the correct sized spanner to undo the nut.
Remove the assembly.
Take the parts to a plumbers-merchants or DIY store.
Show an assistant the parts and ask for advice about replacement washer.
Buy an appropriate replacement washer.
Swap old for new washer.
Replace parts in faucet.
Carefully open isolation valve.