As long as the axial tilt is enough that the shadows of the planets don't cross over each other much, this should work. With the 50 000 km sma example, this would be around 8º of axial tilt minimum (arcsin(R_planets/SMA)) (note: Earth is around three times this). This means eclipses would (somewhat like reality) have two "seasons" per year, when the orbital nodes line up with the sun. The greater the axial tilt, the shorter the eclipse seasons are. During the eclipse seasons, there will be a total eclipse every day, with the exceptions of the first and last days of each season, which are a transitional period with partial eclipses. The centers of each eclipse season on the equator line up with the spring and fall equinoxes. The eclipse seasons themselves are the same length everywhere on the planet, but they don't happen at the same time everywhere. The eclipseless winters are short, and the eclipseless summers are long. At the poles, the entire summer half of the year has no eclipses. At the equator, the eclipseless seasons are the same length.What about a scenario that gives an axial tilt to the barycenter at some point, thereby lessening the frequency of eclipses?
I'm not sure what the climactic effect of a noon eclipse every day in fall and spring would be, but it probably wouldn't be negligible. The eclipses would (again assuming 50 000 km sma of the planets' orbit) last roughly an hour (arcsin(R_planets/SMA) * (hours/day)/180º) at the equator on the equinoxes, and their length over the course of the year would follow a sine wave (with the points where it dips below the axis being summer and winter, when eclipses wouldn't happen) (I don't know the exact equation).
As for the societal impacts, It's easy to imagine the first eclipse marking the new year, especially because this would coincide with the beginning of spring. It would become trivial for early civilizations to know the exact length of their year by timing between the beginnings of eclipse seasons. They would probably notice a discrepancy in their calendars; cities with similar latitude to Athens and Alexandria would see their first eclipses several days apart (assuming a year length similar to that of earth; it could be on the same day if there are few enough days per year). Some calculations from there allow them to determine:
-Their planet is round
-The radius of their planet (likely more accurately than OTL early estimations)
-The radius of their twin planet
-probably some other things too
The inner hemispheres will have much more advanced astronomy. Finding longitude would never be a problem in the inner hemispheres, with or without axial tilt; simply look up.
It wouldn't get very dark at night, seeing as the other planet would reflect quite a bit of light. The darkest times would be midnight around the equinoxes, because the eclipses would block out most light on the other world, so there wouldn't be much to reflect. In winter and summer, there would be no true darkness.
Each planet would probably have much better maps of the other planet than their own, seeing as they could just look up to map the other planet.
Galileo's telescopes had 20x magnification. With around 90 000 km distance between the planetary surfaces at their nearest points, and assuming proper contrast, it should be just about possible to make out features smaller than a km on the other world, which would present some interesting opportunities for civilizations on both worlds.