Why is the conservation of lepton number a thing? Since there is no way to tell neutrinos and antineutrinos apart, does conservation of lepton number make any sense?
As far as we can tell, the neutrinos and antineutrinos are indistinguishable so from the empirical point of view lepton number is not a conserved quantity?
 A: 
Since there is no way to tell neutrinos and antineutrinos apart

There is. Neutrinos are distinguished from antineutrinos since they have opposite signed lepton number and opposite chirality relative to each other. They were also first detected in 1956 as part of an experiment to first detect neutrinos.

does conservation of lepton number make any sense?

Yes. Conservation of lepton occurs in beta decay for example.  In beta decay
$$n \rightarrow p^{+} + e^{-}  + \bar \nu$$
or
$$p^{+} \rightarrow n + e^{+} + \nu$$
where in the first reaction, lepton number before = 0 and after +1 + (-1) and for the second reaction, 0 = -1 + (+1). Clearly conservation of lepton number is followed (and the same for all other interactions involving neutrinos and antineutrinos) and does indeed make sense.

lepton number is not a conserved quantity?

Yes, it most certainly is.
A: Neutrinos and antineutrinos are indistinguishable by most of their qualities, butt not all. One of the quantities that distinguish them is exactly the lepton number, which make them interact with other particles in a different way. For example, a neutrino can take part in the reaction
$$ n + \nu_e \rightarrow p^+ + e^-$$
but antineutrino can't; on the other hand, antineutrino can take part in the reaction
$$ p^+ + \bar\nu_e \rightarrow n + e^+$$
while neutrino can't.
There's also a difference in a property chirality. All neutrinos are left-chiral, and all antineutrinos are right-chiral.
At  least, that's the case in the Standard Model of particles. There are some extensions of SM that allow the neutrino to be its own antiparticle (so neutrino and antineutrino are not only indistinguishable, they are the same particle), and in these extensions the lepton number is indeed not conserved.
