I'm going to have to give you the 1999-2005 timeline of their use. I'm pretty sure things changed from 2006-2017 with new AAs and the like. But pretty much the pet was a persistent DoT. It attacked whatever attacked the SK or whatever the SK sent it after.
Savvy players learned that they could use it to pull mobs with so they didn't get more than one group on them. Really savvy players learned they could use Feign Death in a weird combination (I used to play a necromancer and could do the same thing, but for the life of me I forgot the procedure on how to do it) to pull single mobs from groups without using lulling spells.
Later they could sacrifice the pet for extra health or use it as a center for AOEing spells.
I did that on purpose as I'm not trying to make a new class here. Just show a rough example. Anyway one way a tank pet could work is something along the lines of -insert theme here- thing that draws agro to itself and basically tanks. Its HP limits how long it will last and when it expires, the threat transfers to the user. The benefit is the HP acted like a 'barrier' meaning the tank didn't need healing for as long as the pet was up. The pet doesn't even have to be mobile, if they want to make it a crystal, banner, or whatever.
In this example, the pet doesn't even have to do damage. Just a provoking/taunt effect that holds agro on it long enough to take a hit or two to die/withdraw. Again depending on theme.
Personally I would have any pet that a tank uses to be something that doesn't have 100% uptime. Last thing a tank needs is to manage two characters all the time. That way a rotation involving a pet reminds fluid and intuitive. That's just my personal take though, others may be fine with it.
The great thing about FFXIV is its potency system and modularity. It makes adding new themes (jobs/classes/abilities) really easy and only limited by imaginations. The hardest part is making it fun and interesting.