People say "Dogs must be kept on a lead (leash)" in the dictionary.
5 ATTACHED attached to or hanging from something
She hung her coat on a hook.
Dogs must be kept on a lead (or leash) at all times.
The dictionary also says "It's as if somebody put June Cleaver's pearl necklace and apron on Madonna.", so I reckon "to put a necklace on somebody" is a valid construction, thus I think "to put the necklace on the dog" is valid.
Now, a leash and a necklace look similar because both have a round piece that can be put around the dog neck except that a leash also has a long rope connecting to that round piece for the dog's owner to hold.
Say, now the dog is wearing nothing and you want to prepare for the dog for a walk.
Can we say "to put the leash on the dog" the same way we say "to put the necklace on the dog"?
But the dictionary says "to keep the dog on the leash" so I am not sure if we always have to say "to put the dog on the leash"?
Note: we can say "to put the collar on the dog" but you can hook a rope with a hook to the collar and turn "the collar & the rope" to a thing that can be used like a leash. That is even more confusing.