How long is the 60 days of Game Time Cards?

A 60 day game card is roughly the equivalent of a two month subscription. It will expire exactly 60 days from when you apply the code, regardless of how many actual hours of gameplay you have over those 60 days.

This is common throughout most MMORPGs, including "Final Fantasy XI", the MMORPG prequel to "Final Fantasy XIV".

There are reasons one may favour buying time cards, as opposed to outright paying for a monthly subscription:

• The time card does not have to be used straight away. You can choose to start the 60 days whenever you want. By applying the code immediately before playing, you do not lose the days you might have lost between an automatic subscription payment and the next time you played the game.
• Time cards often do not require a credit card for automatic resubscription. You can prepay your subscription, without worrying about another payment being taken out if you later forget and are not otherwise playing the game.
• As mentioned in the quote, you can buy the time card from a physical retailer. Some people prefer purchasing from a physical retailer, while others may be restricted to it. As a great example, one of my favourite games of all time is "World of Warcraft"; as a sixteen year old child, I did not have access to online payments, and neither did my carer. Even then, there was no way they would allow me to have an online subscription. I was allowed to play "World of Warcraft" because I could save up my money and buy time cards from a physical video game store.
• You can give time cards to someone else; this makes it a lot easier, and a lot more convenient. You don't have to have their account details; you do not even need to tip them off that you're buying them the gift. All you need to do is buy a time card, and give this to them, instead.

Always check the price of the card versus the time it provides and the other subscription options. Often, MMORPG subscriptions are slightly cheaper on the recurring monthly payment model, as opposed to the prepaid time card model. If you can afford it, there are often discounts and further benefits for outright purchasing a yearly subscription, if this is at all available and applicable to your playing habits.

If it is your first time playing the game, there are usually also trials to try the game for free. This allows you to play the game before submitting to a monthly payment, saving you the additional costs if you decide the game isn't for you within the first week. Final Fantasy XIV has a free trial that lasts up to level 60.

Lastly, ask your friends before playing. There may be additional bonuses for signing up friends, and if your friends are playing, they will both know of these bonuses and be interested in signing you up with their refer code. These bonuses can both enhance your starting experience and provide additional time. If you are referred to "World of Warcraft" by a friend, for example, you gain bonuses to make it easier to level when playing with that friend. Your friend also gets free game time, which depending on the circumstance, they may be willing to share.

It's just 60 days, all online subscription games do this.
(Or 30/90 half year or full year)

So if you are not logged in on that day, that day is technically speaking "wasted"

It's 60 real-time days

If you look at the page you quoted in French, the text you quoted is a bit different :

Les cartes prépayées 60 jours sont un moyen de paiement alternatif qui permet aux joueurs de prépayer leur abonnement. Elles sont disponibles pour les versions Windows, Mac et PlayStation®4.

Votre période de jeu de 60 jours commence au moment où vous enregistrez le code de 20 caractères qui se trouve sous le panneau à gratter au verso de la carte prépayée.

The part that is interesting here is the second part, which could be translated to (emphasis mine) :

Your game duration of 60 days begins when you register your 20 characters code located under the scratch panel on the back of the prepaid card.

Adding to this that most MMO nowadays have this subscription system, so it isn't surprising.