The French have their own way of doing things. That goes for pretty much everything French related. Apparently, when it comes to Paris restaurant reservations, French restaurateurs are not big fans of emails. Instead, they prefer phone reservations.
This is one of the least reliable methods because it leaves no trail of your reservation. In other words, you can't prove that you ever made the reservation in case they forget to record it. It pretty much comes down to their professionalism.
So a reasonable fear would be going to a restaurant, only to find out that they don't have any record of your reservation. But generally speaking, French pros won't lose your reservation.
Paris restaurants require that you confirm your reservation a day or two before your intended visit. If you do not, they may either call you to confirm or they may even cancel your reservation without warning and for no other reason. If you call to confirm your reservation and they do find your booking, it is a pretty good indicator that your name will be on their list, when you visit.
You can always give your hotel's concierge your list of reservations and they will take care of any confirmations for you. Some luxurious hotels even have dedicated software for just that. And they give you a printed copy of your confirmation orders.
By the way, very few restaurants in Paris require a credit card when making the reservation.
So there you have it. This is how restaurant reservations work in Paris.