The count of ninety-nine rests on a report in Bukhari and Muslim. The enumerated list does not, and the distinction is well known within Islamic scholarship while being almost universally lost outside it.
The familiar list of ninety-nine specific names comes from a report in Jami at-Tirmidhi, transmitted through al-Walid ibn Muslim. Tirmidhi himself labelled the report gharib, singular, unusual, and noted that the names are not present in the other routes of transmission from Abu Hurayra. The critical assessment that followed, including from Ibn Taymiyya, is that al-Walid took the enumeration from a Syrian teacher and inserted it, and that the list is therefore the work of a scholar rather than the words of the Prophet. Modern gradings commonly mark the enumeration weak.
The consequence is that variant lists exist. Different compilers arrive at different ninety-nines, drawing names from the Quran and from other hadith, and disagreeing over which attributes qualify as names. Ibn Hajar and al-Qurtubi discuss the problem directly. Some scholars hold that the names are not restricted to ninety-nine at all, and that the hadith means only that whoever comprehends ninety-nine of them attains paradise, which would make the list open rather than closed.
None of this is contested between Muslims and outsiders; it is an internal question of hadith criticism with a long literature. It is worth setting out because the ninety-nine names are frequently presented in general reference works as a fixed and revealed list, and the tradition's own scholars are more careful than that. The number is canonical. The particular ninety-nine on any given poster are a compilation.