You can't have two Live connections open at once; the second one will cause the first one to close.
I don't know if multiple concurrent device connections are tracked in other cases or not.
Really isn't much different than having multiple tabs open to the site (such as a Live connection, forum tabs and groups for example). Now, if you are on Live and the Opening and/or Game Explorers at the same time, that may be a different story.
I've found that I can do certain things better on certain devices. For example, I have apps for chess.com on my phone and tablet and I use the website as well. But sending a message to someone is not so easy on my tablet but I can do it on the website easier. But that's not necessarily my issue.
I've tried to avoid opening up the website while I'm playing on my tablet (my favorite medium to use while playing) because I worry that it might alert certain measures that I might be cheating. But just now, I started to play against someone I've had a few games against and I was curious as to what our record is. I just wanted to check the stats while not having to go away from my game on the tablet. It is very convenient to be able to check on the website while a game is going on in my phone or tablet.
What I want to know is this: Is that allowed or will it possibly trigger some kind of mechanism that I might be cheating? I'm not cheating, of course, but I'm worried that having two accounts open at the same time might seem suspicious. Again, I'm only doing that out of convenience.