The rules are quite clear that no magic armor can have a enchantment bonus higher than +5. That does not include the increase from special abilities that count as an enchantment bonus. It only includes the bonus that increases the AC of the armor. The absolute maximum allows for the combined enchantment bonus and special abilities are +10. In order to have a special ability the armor has to have at least a +1 actual enchantment bonus that means the maximum special abilities can be is +9. If the armor has an actual enchantment bonus of higher than +1 the maximum in special abilities is reduced by the increase in enchantment bonus. So, at +2 you can have a maximum of +8, +3 would mean a maximum of +7 etc.
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I am inclined to believe, as a General Discussion topic, that the +5 enhancement limit is not meant to apply to the temporary boost from this ability. Granted, magic weapons use similar but different wording, but I wouldn't stop a bane property from exceeding +5 enhancement either.
The question of Bane Weapons did get a response from James Jacobs in this thread: https://paizo.com/threads/rzs2kjpr?Bane-weapons#1.
I'd assume the same general rule should apply to armor: The listed maximum is just the 'permanent' enchantment bonus max and higher bonuses are possible with conditional effects like Bloodthirsty.
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I posted this in the duplicate thread but...
Probably not. Though arguments can ensue.
There is a FAQ entry that says that weapon enhancement bonus can go above +5 with an ability like bane (which applies only vs. a specific type of enemy) but can't go above +5 with a generic boost that applies to all attacks, like a paladin's divine bond.
There's not an FAQ on armor enhancement, but the same principle should apply. Bloodthirsty is an enhancement bonus to armor vs. all attacks, so it shouldn't allow you to go above +5.
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Also, why would your player really want bloodthirsty? The answer can only really be that they have rage and plan to use it a lot. In that case, sure, they're getting a +3 AC bonus for the cost of +2...
but only when raging and only for one round after landing a melee hit. Otherwise it's only +1 (when they hit). Unless they're built around raging (a lot to make it worth it), and hitting in melee (which is probably a given if the first part is), why not just pay for the +2 bonus and have a +2 AC bonus all the time. That would at least counter the usual –2 penalty from raging before they hit or if they miss (or don't attack one round).
Bloodthirsty is to armor as furious is to weapons. (somewhat)
But yeah, I agree. I don't think this is a good buy. The text of the property makes me suspect that when originally written bloodthirsty had a mechanic like "enhancement bonus increases by +1 each round you make a successful attack" and it got changed in editing.
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Of course, weapons (and presumably armor also) can get at least a +6 enhancement bonus (see the Black Iron Axe in Mythic Adventures), but they are minor or lesser artifacts and not to be able to be created by normal mortal crafters.
I could see a Mythic crafter being able to make such weapons and armor under very specific circumstances like using Torag's Worldforge to forge such a weapon or armor.
You'd have to get Torag's permission to use it and likely have enough Mythic Power to fuel the forge for long enough to forge the item which itself is not an easy task. The DC 46 Fortitude save to survive the drained condition (if you're crafting long enough) is also a bit prohibitive.