Eel has been named in jurisprudence texts as الجريث , الجري ,مارماهي and أنقليس [variant spellings of this also exist]. It is described as a fish in an elongated shape like a snake, with a flattish body. However some opinions also hold that one of these names is used for a fish without scales or Silurus.
Eating it is halal according to the basic principles of the four Sunni madhabs. Because the rule of the majority is that all creatures which live in water are halal, based on the general meaning of the text:
أحل لكم صيد البحر
Lawful to you is game from the sea
— Quran 5:96
And the criterion of the Hanafis is that only the genus of 'fish' is halal from water creatures, and they accept that eel is a type of fish. Their view is based on the hadith:
أحلت لنا ميتتان ... فالحوت والجراد
Two kinds of dead meat have been permitted to us ... fish and locusts
— Ibn Majah
I could not find any reliable indication that any of the mainstream Sunni schools holds eel to be haram. On the contrary, the fiqh manuals imply that there is an extensive agreement that eel is halal:
كالجريث يقال له حية الماء ، وهو على شكل الحية وأكله مباح بالاتفاق
Like the eel, it is called the snake of the water, and its shape is like that of a snake and eating it is permitted with agreement.
— Tafsir al-Baghawy
قيل لأبي عبد الله: يكره الجري؟ قال: لا والله، وكيف لنا بالجري؟ ورخص فيه علي والحسن ومالك والشافعي وأبو ثور، وأصحاب الرأي، وسائر أهل العلم. وقال ابن عباس الجري لا تأكله اليهود. ووافقهم الرافضة، ومخالفتهم صواب
Abu Abdullah (Ahmad ibn Hanbal) was asked whether eel was disliked, and he said: No (...). And Ali, Hassan, Malik, Shafi'i, Abu Thawr, the Ashab al-Ra'y (Hanafis) and other scholars allowed it. And Ibn Abbas said: The eel is not eaten by the Jews. And the Shi'as agreed with them and opposing them is correct.
— Mughni - Ibn Qudamah
ويستوي في حل الأكل جميع أنواع السمك من الجريث والمارماهي وغيرهما ; لأن ما ذكرنا من الدلائل في إباحة السمك لا يفصل بين سمك وسمك إلا ما خص بدليل ، وقد روي عن سيدنا علي وابن عباس رضي الله عنهما إباحة الجريث والسمك الذكر ولم ينقل عن غيرهما خلاف ذلك فيكون إجماعا
And this ruling is equal for all the categories of fish, such as eel. That is because the evidence that has been presented above about the permissibility of fish, does not differentiate between the types of fish, unless there is a specific evidence which makes an exception to it. And it is narrated from Ali and Ibn Abbas that eel is permissible, and opposition to this is not reported from any other sahaba, hence it is a matter of consensus.
— Badaa’i’ al-Sanaa’i’
There are however outlier opinions of individual scholars, for example:
Some have considered eating eel to be makruh because of the story that eels are the remnants of those who were punished through transformation of their physical bodies. This is the reason cited in owari's answer. Ibn ‘Arafah and Ibn Hajar etc. have said that Ibn Habib - the Maliki scholar - held this opinion.
عن علي أن النبي صلى الله عليه وسلم سئل عن المسوخ فقال: هم ثلاثة عشر: الفيل والدب والخنزير والقرد والجريث والضب والوطواط والعقرب والدعموص والعنكبوت والأرنب وسهيل والزهرة ... وأما الجريث فكان ديوثا يدعو الرجال إلى امرأته حليلته
Ali narrated that the Prophet ﷺ was asked about those whose bodies were transformed, and he said: They are thirteen:
elephant, bear, pig, monkey, eel, mastigure, bat, scorpion, da’moos, spider, rabbit, Canopus and Venus.
As for the eel it was a dishonorable (cuckold) man who invited other men to have relations with his own wife
— Kanz al-‘Ummaal (Note: This hadith has been classed as fabricated, see الموضوعات لابن الجوزي )
There is a weak report that Muhammad bin Hassan from the Hanafis considered it impermissible. The basis for this is either disputing whether eels are from the genus of fish, or based on the report that they are the remnants of transformed people (see المغرب في ترتيب المعرب and رد المحتار)
There are weak reports that Ali ibn Abi Talib disliked or did not eat eels (see Musannaf Abdur Razzaq and Musannaf Ibn Abi Shaybah). They do not clearly mention a reason, although one suggests that it was because the People of the Book do not eat it. On the contrary there are also reports that Ali praised it and approved of eating it, or did not see anything wrong in eating it.
You have mentioned seeing vague claims that Maliki or Hanbali schools consider eel to be haram. Indeed there are some websites which claim this, for example https://questionsonislam.com/question/seafood-haram-or-halal says this about the Hanbalis. And a few Hanafi fatwa sites says this about Imam Malik like https://daruliftabirmingham.co.uk/seafood-in-the-four-madhab/.
I could not find any evidence that Imam Malik or the Malikis at large consider eels to be Haram. My guess is that this could be a confusion caused because the fact that according to a report Imam Malik considered خنزير الماء (sea pig) to be makruh (tanzihi). Some scholars hold that those sea animals are haram or makruh whose land counterparts are haram. That is because the wording of the texts which forbids the land animals can also extend to the sea animals because of their similar names. Hence, for example these scholars hold that the animals which are called sea pigs, sea dogs, sea donkeys should not be eaten. I could not find any source which says that Imam Malik extended this to eels. The Hanafi books that daruliftabirmingham.co.uk cites after making this claim do not say this at all, rather Hidayah states that Imam Malik considered all water creatures to be Halal.
I could not find any evidence that the Hanbalis consider eels to be Haram, indeed the quote from Ibn Qudamah that is cited above says explicitly the opposite. However there are other sea 'snakes' (not eels) which can survive in both land and water. These are the ones who are considered haram by the Hanbalis (and for that matter also the Hanafis and one opinion of the Shafi'is). This might have confused the authors of the website. See Kashshaaf al-Qinaa’.