Yes, surface tension can change with the area, but is is important to recognize the ensemble you're in.
Take first the case of lung surfactants, where you have water, air, and the surfactants in between. If you increase the area while keeping the number of surfactants constant, the surfactants become less densely packed, being in a stretched state of sorts. This is to say that the tension increases (as it would if you were stretching an elastic rubber band). Now suppose there were a lot surfactants in the water in micelles (past CMC) and you then increased the area: The tension would not change. When the surfactants become more sparsely spaced, the micelles from the bulk will fuse in to the surface which will take the area per surfactant back to its equilibrium value. Obviously this equilibration process will happen over a certain timescale and is not instantaneous (see Ward & Tordai). Similarly, if you increase the area, but your ensemble is such that the number of surfactants increase with the change of area, you will get no effect on surface tension.
Finally, the case of soap bubbles is slightly different, for there is no obvious bulk reservoir here as we're talking about a spherical air-surfactant-air system. That said, it again depends on the ensemble you're in. If you increase the area by pumping more air into the middle of the bubble, the surfactant area density changes, and then so will the surface tension. But suppose you had a reservoir of surfactants and were to increase the size of the bubble: the tension would remain the same. To address your question of why we can assume soap bubbles have a constant surface tension is, I guess, because the bubble would probably break immediately if you were to stretch it. And there is little energy penalty for doing so. I know there's been quite a bit of interest into how much tension vesicles can take and when MscL opens to release said tension (and the corresponding buildup of internal pressure), so you might follow up on those for more detailed analyses for a similar situation (I'm guessing the water immersed system is somewhat more stable due to the smaller tensions and system sizes). Note, though, that the case is very different for the planar water-air surface in the lung where breaking would expose water and air into direct contact and thus come with a large free energy cost.