There is OpenVR(SteamVR) and OSVR, two open standard API's that are working to do this. But it's difficult. VR headsets are not just 'displays' like a monitor or whatever. Add in different motion controllers and it becomes even more difficult.
Basically, the issue right now is that all of this is in a relatively primitive state. We've been able to get VR 'working' already, but to get it to a level where it's stable and consumer friendly for the average person, while meeting the incredibly high standards of performance and latency and whatnot requires a WHOLE lot of 'new' ways of going about things. It also requires Nvidia, AMD and Microsoft all working, doing their part to implement VR-specific support for API's, drivers and integration with the OS, which has taken a fair bit of time and probably accounts for the main reason we haven't seen consumer headsets out yet. It's also why we shouldn't expect consumer-level support for Linux or Mac anytime soon.
Anyways, needless to say, there is no accepted 'standard' yet. OpenVR and OSVR are certainly working towards that, but doing this can sometimes mean slower progress. Oculus seem to want to move forward as fast as possible and dont feel like compromising their goals just to be part of that push(though they are also contributing indirectly by innovating with new ideas and implementations that can be adapted elsewhere). Personally, I think that's fairly understandable when we're months away from the DEBUT of these consumer headsets into the public domain. HTC said that the reason for their delay was due to a 'big breakthrough', but devs working on Vive games have noted that SteamVR was NOT in a state to be released to the public yet, so I'm willing to bet that was probably a bigger reason for the delay than anything else.
In the end, established and open standards will prevail, but it will take a little while to get there. Until then, there will be *some* either/or sorts of experiences(Oculus SDK only works for Rift), but I think the biggest issue will be that an open standard like OpenVR/SteamVR is unlikely to provide an optimal experience for all users early on. It may work great with the Vive, but maybe not so much with the Rift, for instance. And in VR, something not working as well isn't just about accepting lower performance or whatever, it can ruin the experience and make it unplayable or it might simply not work properly at all.
So we'll see. The current situation is not ideal, but it should definitely improve over time.