TL;DR: Allah says in the Qur'an that mentioning His Name is a requirement, and from Sunnah we learn that the time of this is always during slaughter. Two ahadith are examined below.
According to Qur'an and Sunnah: Every animal has its right and edible animals keep their rights. We're given the permission to consume them in the halal slaughter way, in that case the rights are passed over to us. The animal killed without Allah's name pronounced is maytah and every maytah is haram to eat. This is because these animals will take back their rights in the hereafter if they are eaten.
Firstly, you may eat fish.
Secondly, the correct ruling* is that any animal that was killed without any of Allah's names is haram to eat, even if it was killed according to Kosher rules, which are very similar to our rules (cutting the jugular vein, etc.) This is because in Islam we're not allowed to steal others' rights and unless an animal is killed in accordance with the Shariah, including the pronouncement of Allah's names, the animal will not make himself halal for us, which will increase our torment of grave (because except compliance with Shariah, we're stealing the life of the animal). Unfortunately, saying "Allah" after the death of the animal makes no change to whether it was halal, so by default all meat by People of the Book are haram since I don't know a single non-Muslim butcher who pronounces His Name on meat.
*: I heard this from a modern scholar who is amazing in his knowledge of the four madhahib and even just Islam without the rulings by scholars of madhahib.
EDIT: I learned from the scholar the meaning of the hadith quoted in Rebecca's answer:
Some people said, "O Allah's Messenger (ﷺ)! Meat is brought to us by some people and we are not sure whether the name of Allah has been mentioned on it or not (at the time of slaughtering the animals)." Allah's Messenger (ﷺ) said (to them), "Mention the name of Allah and eat it." -- Sahih al-Bukhari 2057 (sunnah.com) and variants Sahih al-Bukhari 5507 (sunnah.com) and Bulugh al-Maram [grade: hasan] (sunnah.com)
First things first, the meat in question is by default halal. This is because doubt does not overcome certainty and the default state is slaughter in a Muslim region. This would have been clear even if the incident in the hadith didn't take place, because there's no doubt in Islam, but only judging by clear evidence, so as long as there's no evidence, there's no possibility of impermissibility.
Notice a distinct feature of this hadith: When we're in doubt about wudu, we don't say basmala, we just continue. Likewise for any other waswas. But why do we pronounce basmala in this case?
Consider the case that when something is halal for the hereafter but in reality it's not. For example, we forgot a sajda in a prayer but don't notice it, it's perfectly fine in the hereafter. In reality, it's missing a part. Likewise for wudu.
But for this case, if we eat something haram by mistake, then we have najasah in our mouth. Our prayer is valid, because we aren't aware of it, but we have najasah. This is a more serious problem than, for example, missing a part in prayer, because it involves dirtiness.
So why do we say the Basmala in this case?
Because Allah, through His generosity, cleans the najasah when we do dhikr.
You may say "Hold on! This is such an extreme explanation and it's much more likely that pronouncing basmala makes every meat halal!"
I say: The requirement of saying the basmala is clear in Qur'an and Sunnah during the time of slaughter. That's why the hadith above is unique to a case of doubt, as a mustahabb action.
To name one:
It was narrated from Rafi bin Khadij that the Messenger of Allah said:
"If the blood is shed and the name of Allah is mentioned, then eat, unless (it is slaughtered) with teeth or nails."
whereas otherwise it would have been:
"If the blood is shed, then mention the name of Allah and eat, unless (it is slaughtered) with teeth or nails."