I can assure you it's not sonar for people. The signal processing is way more complicated than traditional sonar. Sonar also has number roughly 10 channels. Ultrasound transducers have 10x more. The array is generally 2d and not planar either where sonar is generally a 1d planar array.
I just looked at the Wikipedia page for SAR. I think what you are doing requires a lot more signal processing than an ultrasound. At it's most basic, an ultrasound is really just a 1d graph of signal intensity. You should look at some old ultrasounds, it's pretty obvious how low the resolution is.
I'm sure you know more about signal processing than I do, but trust me when I say a simple diagnostic ultrasound is a pretty rudimentary bit of kit. Most medical imaging is pretty simple actually (not accounting for signal acquisition). Radiologists are trained to read fairly abstract charts, and they want as little processing as possible. Imagine if a CT machine tried lining up images, rather than presenting the raw slices. That might make sense for mapping data, but if you were trying to diagnose a displacement of something, like a broken bone, having the image "fixed" wouldn't do you much good.
That's part of the reason why older ultrasound images of babies are so inscrutable to the casual observer. Since the technician is slowly sweeping a 1d or 2d array by hand, the printed image ends up looking pretty weird if the baby moves. An ultrasound can be 100 db inside the womb[1], so the baby tends to start moving when the ultrasound is performed. The horrible images aren't much of a problem, because the images they give to the parents aren't really used for diagnosis. They use the monitor for that purpose. If there is something the tech wants to explore further, they just look at that area some more.
Based on my limited knowledge of SAR, it seems like the processing is way more important because you are working with data that has been captured in the past.
You can have a basic ultrasound with only on channel. The following patent is from 1985, and has a pretty good overview of the field at the time. It appears that most if not all untrasound transducers at the time were large single channel instruments.
This patent, from 1989, indicates that most ultrasound transducers are either single element or linear arrays. It was the earliest patent I could find with a cursory look that had a 3 dimentional array.
Regardless, even Wikipedia suggests that most of the arrays used for medical imaging use either a single element or a phased array:
To generate a 2D-image, the ultrasonic beam is swept. A transducer may be swept mechanically by rotating or swinging. Or a 1D phased array transducer may be used to sweep the beam electronically. The received data is processed and used to construct the image. The image is then a 2D representation of the slice into the body.
3D images can be generated by acquiring a series of adjacent 2D images. Commonly a specialised probe that mechanically scans a conventional 2D-image transducer is used. However, since the mechanical scanning is slow, it is difficult to make 3D images of moving tissues. Recently, 2D phased array transducers that can sweep the beam in 3D have been developed. These can image faster and can even be used to make live 3D images of a beating heart.
Point being, I think you are wrong. I'm not an EE, so I can't speak towards signal processing, but I am a biologist by training, and I don't see a clear reason why sonar principles wouldn't work. We are basically a bag of salt water.
Also, I am familiar enough with ultrasound to be sure that models with only a single transducer are very common. Hospitals and the like might be using the fancy-pants multi-dimensional arrays now, but the units we used to image things in college were definitely not multi-dimension. For one thing, they were older than the patent that demonstrated multi-dimensional arrays.
