It is unusual to lease rather than loan, but you might do so for say a talented competition horse who was worth paying to lease. That may include some affiliation fees etc.
You might lease a high level competition horse where the ownership is complex; for example where more than one person owns the horse. The terms of a lease are usually more restrictive and include specific competition requirements.
Even with a loan arrangement there may well be an annual fee to loan a good horse. But as with lease you are then expected to meet all running costs etc and the pony is yours 24/7.
With a share you only have the pony on designated days and you may find what you can do with them is restricted. You may or may not have to do chores; you usually do have to pay something toward upkeep.