In ordinary beta decay, an electron and an anti-neutrino, together with a proton, are emitted. The proton has zero lepton number, the electron +1 lepton number, and, it is said, the neutrino-type particle has -1 lepton number, so is an anti-neutrino. Is there any independent experimental reason for deciding that it is an anti-neutrino rather than a "regular" neutrino, or is it termed an anti-neutrino just to make the law of lepton number conservation be true? If the neutrinos from ordinary beta-decay could be made, or were observed to, collide with other neutrinos that were independently known, by some different type of criterion, to be "ordinary" neutrinos, annihilating each other and producing two photons, this would be reason to call the neutrinos from beta-decay "anti-" neutrinos, but as far as I know, this isn't the case. I have read about considerations related to the spin direction of the neutrino relative to its direction of motion, but these seem to be dependent on whether the neutrino is in fact massless, and so travels at the speed of light, which I gather is still uncertain.
There are infinitely many quantities that you could define. Some of them are conserved (or approximately conserved), and hence useful, so we give them names.
For example, we could have defined the "volume" of a cube of water to be the square of its height times the cosine of its length divided by the logarithm of its width. But then it wouldn't stay the same if we poured the water from one container to another. It's much more useful to consider height times length times width, which is why we call that "volume" instead.
Similarly, the number of electrons (counting positrons as negative) plus the number of anti-electron neutrinos isn't remotely conserved, so it's not useful. But the number of electrons plus the number of electron neutrinos is approximately conserved, so we decide to call that "electron number". And summing that over families gives what we call the "lepton number". (This is conserved within the SM to extreme, though not perfect accuracy, with tiny violations due to electroweak sphalerons and (if they exist) Majorana neutrino masses.)
Now you might worry: doesn't this mean the conservation of lepton number is "fake"? Absolutely not, for two reasons. First, within the Standard Model, it's a useful tool to help us know what to expect in experiments. And second, more exotic physics, such the effects of new particles, could change the lepton number, so we can search for such physics by looking for violations of lepton number.
In fact, in a general theory, it's not even guaranteed that you can define any number like lepton number that is conserved. For example, it could have been the case that beta decay can produce an electron and an electron-anti-neutrino half the time, and an electron and an electron-neutrino the other half. Then there would be no reasonable way to define anything like lepton number that's even remotely conserved. It's a nontrivial statement about the Standard Model that you can.
Actually, there is no fundamental reason for lepton number to be conserved. However, the SM Lagrangian respects the total lepton number symmetry. At the moment, there are several experiments, which search for the lepton number violating processes, namely, neutrinoless $2 \beta$ decay.