The inconsistency of accepting both may be obscured by some truly bad arguments for both. For instance, I have seen people argue that DC needs to be a state so that DC residents can have voting rights. Anyone suggesting that they don't is simply ignorant; they have voting rights and elections just like everyone else. I have seen people pull out 'no taxation without representation', somehow overlooking that DC residents do have representation in both the Electoral College and the House of Representatives. What they don't have, as a territory, is specifically representation as a state. Their Congressional representative can't vote on final legislation because DC is not a state; they can debate, participate in committees, and vote in committees. One could well argue that it would make sense to give the delegate full voting privileges, but this need have nothing much to do with statehood -- it's a privilege that we could in principle grant in the House simply on the basis that DC has a large enough population, for instance. What hinges wholly on statehood are (1) sovereignty of statehood and (2) the concomitant Senate representation that follows from it.
On the other side, arguments that the Senate is undemocratic are often quite weak. As I've noted before, elections are not intrinsically democratic, but oligarchical; what we call a 'democratic election' is an election in which special features are added to reduce the oligarchical tendency. The Senate is not less democratic as far as elections go than the House; the elections for the two chambers are pretty similar, and have similar features, with the exception that, unlike Congressional districts, states have permanent borders and are thus far immune to gerrymandering (which is generally seen as undemocratic). So the only thing that can be meant is that the representation is undemocratic. But this is always handwaved; it generally depends on some measure of vote weight, usually implicit and not explicitly defended, and, as I have noted before, there is no single measure of vote weight, and no reason to think that all measures agree. Further, as I've also noted before, one of the ways to make a representative system more representative is to make sure that the people are represented in multiple, different ways; making all the representation the same reduces, rather than increases, the representativeness of the system.
In any case, the DC statehood argument requires that statewise, and thus Senate, representation is important for people's interests to be represented properly, and the anti-Senate argument requires that it not be. Perhaps there's some way to resolve this without contradiction, but I've yet to see anyone discuss it in any way.