Yes, my T and I do it sometimes. I like your way of describing it, SAWE. Better than thinking of it as a staredown, LOL. But sometimes it can go on a long time, and I have been reminded of that childhood game where you looked in the other kid's eyes and the "loser" would be the one who looked away first.
I actually like when T and I hold each other's gaze for a longish period. I feel like it's a bit of an invitation from him to me, "Hey, be here with me. Now! I am here." Like step up to the plate with me.

It makes me feel good. Not alone.
I am very sensitive to the whole looking in each other's eyes thing because my ex-husband was terrible at this. I noticed it the moment I met him. He didn't look at me when I spoke, but off to one side or behind me. It was very unnerving to have someone refuse to look at you look that. It made me feel "not there" for him or beneath his notice or not worthy of his attention or something. And like he could not be trusted (that turned out to be true) because he was so shifty-eyed. After years of living with him, I developed the same habit, because it felt so rejecting to never have one's gaze met, so I started not looking at him too, to avoid the constant rejection. In therapy, I am specifically trying to break that bad habit. I really want to look in T's eyes. I don't want to have become one of those people like my XH (very triggering) who won't meet people's gaze.
So, yes, this whole thing is BIG for me in therapy.
BTW, the thing with the mother's and baby's gaze--I read an article not too long ago that said this is more frequent (although not exclusive) behavior between mother and female babies. Male babies tend not to hold the gaze as long but look all over the room, etc.