Your number of natural attacks is completely independent from BAB. As you note, this is a good thing at low levels. It is often a bad thing at high levels, since you get no iteratives.
The werebear class is hardly unique in this regard. The totemist class from Magic of Incarnum and the psychic warrior class from Expanded Psionics Handbook both have something of a specialty in the area, for example.
And while this is quite good at low levels, when they mean you have more attacks than other people, natural attacks have a lot of drawbacks. For one thing, generally speaking you can’t two-hand them—so no 1½Str to damage, no 2:1 returns on Power Attack. That right there eliminates the optimal way to deal melee damage. And if we compare to dual-wielding, then you have to consider that most people who dual-wield get bonus damage—which the werebear doesn’t—and two-weapon fighting means they too have multiple attacks at 1st. Furthermore, natural weapons cannot be masterwork, and they are difficult to magically enhance, which doesn’t make much different at 1st but can matter quite a bit at low levels above 1st.
And then there is the reality of the werebear class to consider: LA +2, to say nothing of +3, is absolutely brutal, basically crippling. And besides that, you’re locked into 8 levels worth of... not much. The DR is too low to really be all that meaningful, and while the big chunk of Strength and other skills is nice-ish, it really pales in comparison to actual class features, which the werebear entirely lacks. Maybe it looks pretty good compared to a single-classed fighter, but single-classed fighters are also pretty badly off. Compared to the whole run of 3.5e classes, the werebear is decidedly on the low end, three 1st-level attacks or not.