The Messiah according to the Jewish thought was and is (discussed with Jesus also in mind)
1. A descendant of David. The Gospel of Matthew, which is the second earliest Gospel to be written cites Jesus as the son of David but it was probably written 3 decades or more after Jesus died so maybe the Jews never knew Jesus as such. And if Matthew picked up this allusion from an older source at that time there were many people claiming to be "The Messiah" so maybe it was lost in the clutter.
2. The messiah will attain sovereignty over Israel. Jesus entered Jerusalem after a long ministry which the Romans did not find as threatening enough until he did, and within a few days Jesus and his movement was crushed and he was crucified. So he actually did not gain sovereignty over Israel. Thus, the Jews did not recognise him as there Messiah that they were expecting.
3. The Jewish Messiah was to bring global peace, bring all the Jews from all over the world and make them all obey and follow the Mosaic "law" (Torah) which never happened in their eyes.
4. When the Messiah comes, that will be times when there is no one that can be called rich or poor, which is interpreted as all will be rich so there won't be any need or opportunity for charity. So everyone will be equal. The Babylonian Talmud is explicit in that. So that kind of time has not yet come.
5. The talmud also instructs the one who is awaiting the Messiah not to drink wine during the weekdays but only on the Sabbath day or on a special day like a festival. After all, the Messiah can come any day. But this is dedicated to a Nazarite who awaits the coming of the Messiah. While discussing this, it mentions Elijah, the preparer of the path for the Messiah, who will never come on the Sabbath. And there is an argument there that The messiah might come directly without the coming of Elijah but this was all to do with how much the Nazarite was allowed to drink and may as well be a sacred way of warning alone. If John the Baptist asked Jesus if he was Elijah, and he said he is not, (Among saying he is not "that prophet") John is not Elijah either.
6. The war of the Gog and Magog has to take place prior to the Messiahs advent. Gog and Magog has been associated with Alexander based on the Quranic interpretation of the word dhul-qarnain is the dual plural of Qarana which in its natural arabic meaning, means man of two generations thus him being Alexander is an assumption based on the wall story which may have influenced this thinking. So its more of an assumption based on some parallels in Alexander Romance, than based on any kind of real evidence. various theories exist what this god and Magog is, some based on characteristics and some based on the name itself like the Lidian king 7 centuries before Jesus who had a name that sounds like it (Gygas to my memory) and then the Jewish Midrash claiming both refer to the same conquering nation that will wage war against Israel itself. So there are various ideas about this and it has been of paramount importance to many.
7. The advent of the Messiah will happen right after this and it will be the "Worst period for the people of Israel". One must remember that the Messiahs job is to bring all the people of Israel from all over the world and get them under the Mosaic Torah. So when we refer to the "people of Israel" it does not mean those reside in Israel which is a big misunderstanding of those who are non-jewish. Thus taking this criteria into consideration the persecution of the Nazi regime seem to be the "worst period for the people of Israel". This is mentioned as I remember in the same area where the singing of praise or the small "Hallel" (Hallelujah is made of the word Hallel which means to fanatically praise)
Thinking from the perspective of the Quran
8. the Messiah will never be a God because God is ahadun, there is nothing else that is divine (no ilah), and of course he already came and his name is eesa. Ain, ye, seen. (very similar to the Gree Iesous). He was Bikalimathun Minha or a word from him (which is a synonym for an "ordain of God"), A rasool, Eesabnu Maryama or son of Mary, he was born while Mary was guarding her Farj or chastity. He is named by name, not inference. The return and coming back of the Messiah is not in the Quran (Some people are definitely gonna get hung up on this particular point for a long time I assure anyone who reads this because this is the single point that is paramount to people and their respective theologies). There is no mention of Jesus coming back in the Quran. It is extra Quranic, eisegesis, inference. Anyway, this is what the Quran says about the Messiah. So he has already come and gone (according to the Quran).
Nah, the Baha'u'llah is just another one of a gazillion false messiahs. To be honest, I think bar Kochba is a better candidate, and he wasn't the messiah either.