You can technically only specialize in 3 of them in XIV. That said specializing is also kinda pointless(it makes stuff easier to make, but they don't really have spec exclusive recipes anymore so eh).
The reason it doesn't fuck the economy mostly is because crafting takes a lot more time/money/effort than it does in wow(at least back when I played which was like 10+years ago but I doubt they made crafting actually engaging or long to level, since this isn't really wow stuff). For example leveling a crafter to cap from scratch right now requires to level every craft(all 8 or 9 of them) to at least 50ish for cross class skills, which takes both a long time investment and money, even with leves, and it still takes hours of crafting, a lot of which has to be done with at least some presence or 3rd party macro tools. You don't just gather x pearls and y metal and then make 15 pearl necklaces by pressing craft, you have to make each individual items using a skill based(as in you use skills, not that it requires skill) minigame and you need to make everything well enough that it's HQ to turn it for quests and what not, so overall you're looking at ~2mins of crafting per item. Because of the RNG elements in the minigame, macroing it unattended generally requires poorer results, meaning it takes longer/costs more money.
Because of all this, there's not many people crafting at all. The few that are crafting also don't put enough effort to really get to the top of crafting with every classes leveled to cap. On top of that, even just leveling isn't the end of crafting, you then need to gear your crafters, which again require money and time, with a tier system of crafting gear similar to raiding gear. You can only craft meh stuff at first with your vendor gear so you make small amounts of money, and then you reinvest that money into buying materials to craft some better gear so you can craft better stuff and so on. And then you have to meld your gear which is yet another massive money sink, often costing more than the gear itself and you have to redo that every few patches when a new tier of crafting gear is released.
So while you can level all crafting, very few people actually bother leveling all crafts AND gear all crafts. If you do, you make stupid amounts of money as a result compared to everything else(a non crafter non gatherer might make say 200-300k in a 4hours day farming, while a crafter can make 1-2million an hour, not necessarily accurate numbers depends on patch and stuff but to give an idea of the difference).
Wow is an entirely different beast when it comes to crafting so I don't think comparing it to how FFXIV does thing isn't really realistic.