The gemara[1] says that on the day before Tisha B’Av, one should not eat meat. The Shulchan Aruch[2] rules that this refers only to the seuda hamafseket.[3] Clearly, then, the gemara assumes there was no prohibition to eat meat during the Nine Days.
However, there is an old minhag forbidding one from eating meat earlier than that. The Tur[4] says that this is a minhag mentioned in the Talmud Yerushalmi (according to one version) as starting on the first day of the month of Av. The Shulchan Aruch[5] cites various opinions regarding when the minhag begins: the week in which Tisha B’Av falls, the beginning of the month, or from the fast of the 17th of Tammuz (i.e., the Three Weeks). The Mishna Berura[6] notes that the minhag of Ashkenazim is from the beginning of Av. The practice of some Sephardim is to refrain from meat only on the week of Tisha B’Av. However, most Sephardi poskim rule to start from the beginning of the month. In practice, this means to refrain from meat from the second day of Av, because they hold that the prohibition does not apply on Rosh Chodesh, which is normally a festive day.[7]
Your question is whether the above is a fully binding halacha or whether it is a weaker minhag. This is difficult to answer clearly. The long answer, which can be asked regarding many practices, is too extensive for this forum. The short answer is that it is both. Once a minhag develops and is accepted broadly throughout a community,[8] it becomes incorporated in Halacha and is binding on the community, or even in the entire Jewish world.
Note the Aruch HaShulchan’s[9] comments regarding your question: “Our fathers already accepted hundreds of years ago not to eat meat or drink wine from Rosh Chodesh Av until after Tisha B’Av, except on Shabbat. This is to remember the sacrifices and the libations that were discontinued due to our sins … Now, in our great sins, how people take this prohibition lightly! First of all, they violate a Torah law by abusing a vow, for since our fathers accepted this minhag, it is an oath of the Nation of Israel. Besides this, how can we not be embarrassed?! … A nation about which it was said ‘you shall be holy’ will not agree to restrain themselves eight days a year as a remembrance of our house of holiness and grandeur …” Granted that this is a notably strong stand both on the halachic status of a minhag in general, and on this issue in particular. However, it highlights the idea that the assumption that a minhag by its nature is largely unimportant is not simple, especially when the minhag is old and established.
The relative lower expectations regarding minhagim are related to the common situations in which the minhagim were initially accepted and the leniencies incorporated into them (explicitly or by means of historical development). The minhag/prohibition not to eat meat during the Nine Days is an example of this phenomenon, as its leniencies (or lenient opinions in its regard) likely stem from the fact that it is a minhag, not a regular prohibition. We will now present a partial list of such leniencies.
Whereas one may not feed non-kosher food to even an infant, during the Nine Days, he may give meat or wine to a child who is too young to understand the meaning of mourning for Jerusalem,[10] at least when this is needed to fulfill a mitzva.[11] One who is (even slightly[12]) sick may eat meat during the Nine Days,[13] if there is a health benefit from eating meat. One may make a seudat mitzva[14] with meat, including a meal upon finishing a tractate of Talmud (siyum).[15] Regarding the parameters of when it is permissible to make such a siyum and who can be invited, it is notable that the custom has developed to be more lenient than the classical poskim envisioned. We feel that in general, it is proper – when possible, appropriate, and logical – to follow legitimate minhagim as they develop, whether for stringency or leniency. Here, again, the general development has been for stringency, but within some of its specific applications, there is a direction of leniency. This type of balancing is part of the halachic system, including regarding halachic minhagim.