Both Shiah and Sunni traditionalists have reported traditions to the effect in their books. There are more than 399 narrations for the same. For the benefit of my readers, a few of these books are given below:
1.   Yanabiul Mawaddah, page 433
2.   Nurul Absaar by Shiblanji, page 134
3.   Muntakhab Kanzul Ummal, volume 6, page 32
4.   Al Burhan Fee Alamaate' Mahdi Aakheruz Zaman, chapter 2
5.   Musnad Ahmed Ibne Hambal, volume 2, page 133
6.   Sunan Abi Dawood, volume 6, page 94, tradition 4242
7.   Mo'aalime' Sonan, volume 4, page 336
8.   Mustadrakul Sahihayn by Hakim Nishapuri, volume 4, page 466
9.   Hilyatul Awliya, volume 5, page 159
10. Iqdud Durar, chapter 4, volume 1, page 4905
11. Jam'ul Jawameh, volume 1, page 545
12. Al Burhan by Muttaqi Hindi, page 103, chapter 4, tradition 3
13. Kanzul Ummal, volume 14, page 269, tradition 37675
14. Tazkerah by Qutbi, volume 2, page 800
15. Arful Wardi, Suyuti, volume 2, page 65
16. Durrul Mansoor, Suyuti, volume 2, page 57
17. Sawa'equl Mohreqah, Ibne Hajar Makki, page 163, chapter 11
18. Muntakhabul Asar, chapter 2, page 442
19. Al Malahim, Sayyed Ibne Tawoos, chapter 9, page 22-23
20. Basharatul Mustafa, page 250
21. Seratul Mustaqim, volume 2, page 242, chapter 11
22. Isbatul Huda, volume 3, page 607, chapter 32, tradition 120
23. Ghayatul Maraam, page 697, chapter 141, tradition 54
24. Beharul Anwar, volume 51, page 365
25. Manaqebe' Shahre Ashob, volume 1, page 292

His name was not that of the Holy Prophet (saw). The Prophet's name was Mohammed and not Ali Mohammed.

His kuniyat was not that of the Holy Prophet. In fact there is no record of his kuniyat at all.

He was not a Sayyed, but a Mirza. The title of Sayyed is given to a person whose father is from the lineage of the Holy Prophet (saw). Mirza is a title given to a person whose mother (and not father) is from the lineage of the Holy Prophet (saw). However there is a practice in Iran to call respected people as Sayyed which is why the Bab got the title of Sayyed. So in effect, the title of Sayyed applicable to the Bab stems from culture as against birth.

Mirza Ali Mohammed is not from the Ahle Bayt (family) of the Prophet. The Ahle Bayt is a specific term given to the family of the prophet. Even people who are from the lineage of the Prophet (Sayyed, as explained above) are not included in the Ahle Bayt.