Do editors know how many papers I am refereeing on the same platform? I have very recently returned two referee reports for two different Elsevier journals. Within two days, I have received two new requests from the exact same two journals (but from different editors).

Of course this might be a coincidence, but now naturally I am wondering: did these editors somehow know that I had just 'freed myself' from the previous commitments and was again available as a reviewer? Is there, for instance, a field in Elsevier's online platform that tells editors how many assignments I have at a given time?
I think there are two aspects to your question:

1) Do editors see how many other papers you are currently reviewing? 

Yes, in fact this feature is supported by at least some submission systems. I am not sure about Elsevier, but below is a screenshot from Manuscript Central (also known as ScholarOne):



The second and third column list how many open reviews the person has, and when (s)he last submitted a review.

2) Do editors care? Are you more likely to get new reviews when you have submitted your current ones? 

In general, they will likely not care too much. At least in CS there are myriads of different systems in use, and knowing that a person has currently no open reviews in any given one of them tells me close to nothing about what her or his total review backlog looks like. Further, it is not my place as an editor to decide whether the person can take on another review - I can invite her or him, and they will just decide for themselves. That's what the Accept / Refuse buttons are for, after all.