*with the tidal bulge always in the same place, it seems like strong tidal forces would pull the planets into a weird egg-shape?
Technically speaking probably even the solid parts of the planets would be slightly prolate spheroids, though it might take some measuring and some technology for the inhabitants to prove that.
But the tidal bulge would mostly affect the oceans; and if you include the liquid part in the shape of the planet, they would be even more prolate, probably noticeably so even to earlier technologies.
If you include the atmosphere too, that would also be more prolate than just the solid and liquid part.
Or at least that's my guess.
The entire two-planet ensemble would rotate around their joint center-of-gravity in a time-period probably longer than an Earth-type day, and probably more comparable to an Earth-Moon lunar month. IMO that should be quite "inhabitable", since the polar zones on Earth have at times been quite well-inhabited, and they have six-month days and six-month nights.
The process by which the two planets become tide-locked to each other would slow each one's original independent rotation about its own axis, while moving the planets further out from each other. In other words, energy would get pulled out of the spinning of the individual planets, and moved instead into speeding up the linear speed of their orbits around their joint center-of-gravity, thus pushing each one up into a higher orbit relative to that joint c-of-g, in other words, moving them apart. (Their angular velocity would slow down, though.)
Inhabitants might find fossil evidence of life-forms that lived on the planets just before tide-lock; say, maybe, when one of the planets rotated twice on its own axis for each time the two planets revolved about their joint c-of-g. (Or not.)
After one of them is tide-locked, whichever of its sides faces its partner, will never or at least rarely have an entire day without at least a partial eclipse of its sun; at least in its equatorial and tropical regions, and maybe its temperate regions as well. This is bound to affect life. Finding evidence that at one point life there was adapted to having full sunlight for half-a-"month" every other "month" (or whatever the schedule would be like, it would take care and math to work it out), would prove that life evolved before the tide-lock was complete.
The fact that the planets would be
, with their long axes pointing directly toward and directly away from each other; together with the fact that they would be tide-locked; might make something like an "east pole" and a "west pole" more significant than a "south pole" and a "north pole". Though perhaps they would name them "eclipsed pole" and "high-noon pole" or something like that.