No, you may only rent one car at a time. You are, therefore, also unable to have multiple pending trip requests out for different vehicles with times that overlap.
If you need to book another car for a different date, you can rent that car as long as there is no overlap in trips.
When you’ve found the car you want to rent, request it. The trip will be automatically booked when the owner confirms it. If the owner has not confirmed the trip within 24 hours, has declined the trip, or you have withdrawn your trip request before the owner has confirmed, you may immediately request a different car.