Rules as written, it is left to DM or the table for interpretation
Rules as written what you're supposed to do is look at Nondetection and Portent and decide for your table how it works. You're supposed to look at what the intent of Nondetection and the intent of Portent is supposed to be, which means you'll have to read and decide for yourself based on how you understand the rules and what works best for the fun and enjoyment of your table. That is 5e's rules philosophy.
I don't buy the "It doesn't say it is or it isn't, therefore it isn't." You can just as easily say, "It doesn't say it is or it isn't, therefore it is." Lack of evidence doesn't prove anything. The game rules do not claim to be comprehensive, and this is the natural consequence of that. This is the entire point the "rulings not rules" idiom is making. There is no burden of proof on Portent or Nondetection or the PHB as a whole to provide an answer. While that line of reasoning was implicit to 3e and 4e -- both editions that sought comprehensive and complete rules sets -- 5e actively does not do that. 5e D&D is often intentionally vague so as to leave rules open for interpretation and the system intentionally doesn't use keywords or raw mechanics. It does this so that players and DMs have the explicit freedom to interpret the rules for themselves and do what makes sense for that interaction. The designers know they can't possibly foresee every interaction in the rules, so they no longer try.
There is no clear answer by design. Yes, this means that asking, "What is RAW?" on RPG SE for 5e D&D is often a pointless exercise because the answer you should often get is, "RAW it depends on your table." This is why there are so many conflicting answers on RPG SE for 5e questions and why Sage Advice contradicts itself so often.
The rules are less important than the game. What Mearls and Crawford want is for players and DMs to stop asking WotC how to play and just make a decisions and play for themselves. If you make a mistake, admit it and make a correction. It's no less destructive than doing nothing until WotC makes a decision and then maybe changes their mind later.
You're expected to look at whole picture that the rules are giving you and to make a judgement call on what feels the most consistent and correct for your table. Is it reasonable for Nondetection to block Portent? Sure, it almost certainly is divination magic given that it's an ability of the Diviner subclass. How about a Ranger's Primeval Awareness? Well, that works like a spell, even consuming a spell slot, and it would have to be divination magic given the distance, so sure. How about a Paladin's Divine Sense? Hm... possibly, it's pretty close to Detect Good and Evil, but it's really described as the Paladin's senses. A Warlock's Devil's Sight? Hrm, hard to say. It's got elements that only True Seeing can accomplish, and Nondetection probably blocks True Seeing, but it's basically an improved Darkvision spell and that's not even Divination. So maybe partially? A Barbarian's Feral Instinct? Eh, that doesn't seem right, it's not magical. A Rogue's Blindsense? Yeah, probably not unless Blindsense is supposed to be magical, but I don't get that impression.
It would prevent the spell from targeting you
Consider the wording of the amulet:
Whilst wearing this amulet you are hidden from [school] magic. You can't be targeted by such magic [...]
If we were to replace [school] with evocation, it'd be pretty clear by the description that you can't be targeted by evocation magic. I think it's equally clear in the case of divination.
Additionally, this amulet is of uncommon rarity. Blocking damage from a narrow range of spells is not unusually powerful for an uncommon magic item. A few examples of uncommon magic items that block spells:
- Brooch of Shielding: This gives resistance to force damage, and entirely negates Magic Missile (which is, in my experience, much more common than Mind Spike).
- Periapt of Health: This amulet blocks any spells that cause a disease in the target.
- Necklace of Adaptation: This necklace gives advantage to saving throws made against harmful gases such as Cloudkill and Stinking Cloud effects, as well as inhaled poison and some dragon breath weapons.
All of those effects are much stronger than Mind Spike.
Best Answer
Detect Magic would not work on the creature, but may work on items on that the creature is wearing/carrying
Detecting magic on the creature wearing it
The quote you provided for the Amulet describes what it does - and it makes any magic currently active on the wearer hidden from divination magic.
Detecting magic on the items carried by a creature
At first pass, it does seem like detect magic should work here. But I think this is going to be up to some DM decision-making. Most of the time, items carried by a character are often considered part of their character (for instance, magic that causes fire generally stipulates that it won't burn things carried or worn.) However, that is a specific rule and may not be applicable at all times.
It's also important to note (thanks miniman!) that in order for detect magic to detect, the item must be visible. If it's not visible, then the spell definitely won't detect it.
In general, it does seem like divination magic won't hide magical items carried/worn by a character. But as with the fire example above, I think there is a case that it will hide those items. I think it's more fun that someone wearing this amulet is basically covering themselves and everything they've got in this magical protection.
Your table, your rules
The only thing that really matters is that the spell doesn't work to detect items worn or carried, then that's fair to rule, but just remember to be fair, fun, and consistent if this sort of thing happens again.