It's a matter of style, but that style is informed by several things, among them one's emotional health as well as general attitude toward the event in progress.
Yes, Queen Mary also showed a similar style. Her attitude, however, was different. It was obvious that, like the Tsarina, being out in public was not "her thing." (As opposed to say, her charismatic mother-in-law.) But instead of looking strained Mary seems to match the onlookers, stare for stare. When Alexandra was able to muster an imperious attitude she gave off an air of discomfort; Mary's imperiousness, on the other hand, came from a stronger inner core and was respected.
I think that whenever publically on view Alexandra generally did what she could, within her own personality limitations. I've frequently had to speak in public and making direct eye contact is a skill that I'm still working on; it's difficult to look out upon a crowd and see individuals rather than a sea of faces. The "head bobbing" observed on that film is, I think, related to that. How many of us can feel absolutely comfortable being under the scrutiny of hundreds, maybe thousands, of potential critics? Queen Marie of Romania didn't have a problem with such a scenario, but she also had exceptionally strong self-confidence plus a desire to be on exhibit. (Something of the Madonna of her time, if you will!)