There is no restriction against a short-term rental.
And I am not aware of anything the indicates a lease is a form of ownership [Edit: well, REALLY long term leases like 30+ years may have a form of ownership]. "Rent" and "lease" are the same thing, but "lease" is often used for longer-term rental situations.
For the original poster, I'm not sure who "the company" is, so I can't comment on that (my first impression is his client is paying for a leased vehicle and letting him use it, in which case I agree he can't use the Standard Mileage Rate).
