Yup, this is right. Everyone plays on the same servers regardless of the platform you're on, but any expansion you purchase is tied to the platform (so Mac, PC, PSN and soon presumably Xbox). So if you personally want to play on multiple platforms you need to buy multiple copies of the game.

However that only applies to the initial purchase. FFXIV being an MMO still has a monthly subscription fee (unless you're on the free trial, which isn't time limited and includes the base game plus the first two expansions), but you only pay that once a month per account, regardless of how many platforms you play on.

Pretty much everything is stored on the server as well, so regardless of the platform you log into, your character will be the same. There's a small number of things that are kept client side (things like your gear sets and your UI setup), however there is an option to upload your settings from one client and download them on another (so for me for example, my PS5 is my primary device but I sometimes play on PC, so I've copied my PS5 settings to the server, and then downloaded them onto my PC, all through the game client).