Liquid Amps

How about the term "surgically romantic" published in TAS (1976) by that doyen of the High End - Harry Pearson! He was reviewing and describing James Bongiorno's Great American Sound Ampzilla! This was THE amp to own! Did it sound "liquid"? NO! It sounded as clear as a just-Windexed pane of glass! IT SOUNDED BEAUTIFUL! And that was James's intention: to produce a SS amp with the smoothness of the best tube amps and better transient response than the SS competition! And the ability to drive ANY speaker with full power! Ampzilla did indeed set the bar really high! 

Bongiorno was criticized by The Audio Critic for producing amps and preamps that made all sources sound too beautiful! So I think that the term "liquid" can apply to these now classic SS products from GAS!

Disclaimer: I presently own a Thoebe preamp, two Sons of Ampzilla and a Grandson! Classics indeed!

What is the most “liquid” sounding solid state amp and is “liquid” even a thing?

It is, and its a description of how the amplifier makes distortion, or more to the point, what distortions it **doesn't** make.
Solid state amps have a long standing reputation for sounding brighter and harsher than the actual music, this despite having typically very low distortion overall.
The problem is that the distortion they *do* have is entirely that of types to which the ear is keenly sensitive, moreso in most cases then excellent test equipment. This is because the distortions in question are higher ordered harmonics (5th and above) which the human ear uses to sense sound pressure. You probably don't have to imagine how important it is to be able to tell how loud a sound is! Amounts of 0.005% are thus audible.

For this reason, and also due to the distribution of harmonic structure in musical instruments (which is not a coincidence), the tiny amount of distortion most solid state amps make is actually really audible, hence your opening question!
Some of this is caused by non-linearities in the solid state devices, and some is caused by the application of loop negative feedback, which is used to suppress the distortion of the former. Its a bit of a carrot on a stick: no matter how much feedback is applied, the actual use of feedback will generate its own higher ordered harmonics.
This fact has been known for at least the last 65 years (for those interested, see the writings of Norman Crowhurst)!
Generally speaking, tube amplifiers don't make as many of the higher ordered harmonics and so sound smoother as a result.
But in the late 1970s, Sony developed a new kind of power transistor called a Static Induction Transistor (SIT), also known as a VFET. This transistor had linearity similar to a triode power tube and even had soft clipping characteristics. Unfortunately, Sony, like they were with so many of their innovations, was very precious about the use of their technology, and the only amps built for quite some time were their own VFET amps, which were not particularly reliable. They went by the wayside.
However Nelson Pass took an interest in the device and even had some made at one point, although the semiconductor house that made the SITs for him later went out of business. Because of the rarity of the devices, Nelson never made a production amplifier with them, but he did make a kit, and by all accounts it is easily the amplifier you are looking for- it has the least 'transistor' artifact of any solid state amp I've heard.
The amps don't make a lot of power (about 20 watts), and the kit is not being offered to the best of my knowledge, but you might be able to find one that didn't get assembled that is for sale. It would not be that hard to find someone to build it.

Here is excerpt from Absolute Sound Paul Seydor review of my amplifier Benchmark AHB2:

Because accuracy allied to absolutely reliable performance is the goal of all the Benchmarks, they are not products that tend to attract cults or other sorts of starry-eyed enthusiasts, wholly lacking any of the quirks, foibles, idiosyncrasies, sonic flavorings, euphonic distortions, and so on that characterize the objects of most audio cults. Professionals buy Benchmark because they know the products work and are reliable and accurate—indeed, reference caliber. Music lovers buy them because they are neutral and accurate and thus reproduce the tonal character of voices and instruments correctly (and also, I presume, because they are reasonably priced, most musicians, like most other people, being typically not wealthy). But audiophiles? Well, the longer I’m in this racket, the less I sometimes think I understand what audiophiles really want except that a lot of dallying about with components, equipment swapping, and coloration matching seems to be what amuses them. I’m not sure I can in good conscience recommend this amplifier to them as I am not sure they are in search of what it offers: a precision instrument designed to perform the precisely defined task of reproducing music and sound accurately, which it does essentially to perfection. But to anyone else, the AHB2 gets as high, enthusiastic, and confident a thumbs up as my arm is capable of reaching.

AHB2 employs different technique of correcting transistor nonlinearities.  Instead of applying negative feedback to the input it utilizes two parallel amplifiers for each channel - one for the signal and another for the error (8 power transistors per channel).  Correction of nonlinearities is done at the output instead of the input.  This technology called AAA (Achromatic Audio Amplifier) was bought by Benchmark from THX (George Lucas).  To me this amp is as smooth/liquid as it gets, with wonderful extension on both ends and serious slam.

Ralph, reducing bandwidth at the input to one that amp had before feedback was applied should prevent TIM, but requires fast and linear transistors that are expensive.  Using more feedback instead is very tempting since even shallow 20dB feedback improves everything tenfold (bandwidth, output impedance, THD, IMD).