The event's meaning is not the same on this side of the curtain. All but mandatory attendance (sometimes with people literally checking names off a list), officials' speeches and military parades tend to sour the mood somewhat, and, from what I can tell, that memory lingers still. These days, many people just use it as an extra day off.
They don't call it that and don't behave like that, so no. They have anti-NATO, anti-USA and pro-Russia banners (and one pro-China). But there was a march earlier, at another place. I believe the anti-communists were there as well.
I was born in a communist country, you don't need to tell me. There was Labor day march elsewhere, as I wrote in the previous comment. This was a protest. They did not move and actually did call themselves a protest, not march (if you asked about that, they pointed you to the march that was elsewhere and also at another time later in the day).
All right, but I was saying so from the perspective and for the potential benefit of US readers, who celebrate Labor Day in the fall.
As for your other point, at least here in Switzerland, the delineation between "march" and "protest" is not so clear, and May Day usually features both.
