The producer has explicitly stated they are related in the same way the Twilight Zone or Outer Limits episodes are related. They are two episodes of an Anthology. There is no direct relation between the two movies other than production and some meta themes. Parallel universe at best. http://www.latimes.com/entertainment/herocomplex/la-et-hc-jj-abrams-cloverfield-movie-universe-20160309-story.html
The secretive drama isn’t a sequel to the “Cloverfield” movie but rather an extension of a fictional universe. “We do have a big, fun idea that hopefully will get a shot to realize,” Abrams said over the phone. “But in the meantime, I think the easiest way to consider this is that it is something of an anthology. But it's also something else that we're playing with. Fingers crossed that we get to execute.”
When asked if what Abrams was trying to create was something in the vein of “Twilight Zone” or the “Amazing Stories” series but for film, Abrams confirmed, “It's an easy way to understand it at the moment, that it's just a cool anthology. And I think that this movie stands alone. I think [the director] Dan Trachtenberg did a terrific job on its own. But it does have a connection to something that we'd love to see through. But certainly the ‘Twilight Zone’ is one of my favorite shows and if this can be seen in that light, it would be a wonderful comparison…
Again, it's sort of an easy way to say it's an anthology of cool genre films could all fall under the ‘Cloverfield’ banner. But seeing that this is just the second movie, I don't want to be presumptuous. There is a larger idea that we're working on that would be really fun if we could see through. Hopefully that can happen, but in the meantime, the ‘Twilight Zone’ comparison is flattering and hopefully appropriate one.”
The Cloverfield monster was a Godzilla expy, either a natural monster or created by nuclear testing, while Cloverfield Lane monster was a crazy guy, which coincidentally happened to attack when aliens attacked.