In a world without Physical Levels, how would stat assignment and bonuses work?

Well, each Armory System Class trains you in the use of certain weapons, tools, and magic. The more you practice and use those weapons and tools, the better you get at using them, and the more things you can do with them (i.e. gain abilities). But also your underlying physical and mental attributes that help you use those tools and weapons should also improve ... based on which weapon or tool you are ranking up in.

So an Olympic javalin thrower, as he gets better and more practiced, would gain both in strength (to throw farther) and in dexterity (to throw accurately). Meanwhile, an Olympic boxer would gain in dexterity (to dodge and land punches) and vitality (to take punches when they do land). In this scenario, the boxer would become tougher than the javalin thrower, but the javalin thrower would become stronger than the boxer over time. This hearkens back to Hiromich Tanaka's original vision of "organic growth".

So how do we do this in FFXIV?

Whenever a character gains an Armory Class rank, that character gains points in corresponding stats appropriate for the weapon/class he's gaining rank in ... let's say 6 points. Below is possible way of distributing those points.

Gladiator: VIT +3, DEX+2, STR +1
Pugilist: DEX +3, VIT +2, STR +1
Marauder: STR +3, VIT +2, DEX +1
Archer: DEX +3, STR +2, VIT +1
Lancer: STR +2, DEX+2, VIT+2

Conjurer: INT +3, MND +2, PIE +1
Thaumaturge: PIE +3, MND +2, INT +1

Disciples Of Hand & Land would also get stat points as they rank up, based on their class primary and secondary stats (official list here), but it would only be 3 points.

Alchemist: INT +2, PIE+1
Armorer: VIT +2, STR +1
Blacksmith: STR +2, MND +1
Carpenter: VIT +2, DEX +1
Culinarian: MND +2, PIE +1
Goldsmith: DEX +2, INT +1
Leatherworker: VIT +2, INT +1
Weaver: DEX +2, MND +1

Botanist: STR +2, INT+1
Miner: VIT +2, MND +1
Fisher: DEX +2, PIE +1

In any case, whenever a character would gain rank, they would also get 1 point that they can assign to any stat regardless of what class they gained the rank in. Call it a discretionary point, an extra-curricular activity point, or a customization point ... whatever you want. It's to allow some degree of individualization so that every naked Rank 50 Lancer doesn't have identical stats.

A lot of players seemed worried about how stats would work in a world without physical levels. This is one way. And it's not a bad way, in my opinion.