There is no such thing as a "soul" in an amp, it's just physics, the type of circuit, and specs of components. An amp is an amp is an amp.
Now, there's a fundamental difference between a tube and an SS amp. More precisely, tube amps are current drivers with no or limited negative feedback, while SS amps are voltage drivers with 100% negative feedback. This is what makes all the difference, aside from power amp distortion, compression etc, which, I assume, is dialed in anyway.
Each speaker and each type of cabinet has a unique impedance curve. It has a frequency around which its impedance rises sharply (there's a bump), and a frequency after which it rises gradually. When a cabinet is powered by a current driver (tube amp), there's a resonance, and output goes up for those frequencies. When it is powered by a voltage driver, the output goes down. The negative feedback in an amp dampens the resonances (in the case of SS amps, completely), so there output is flat. Modern speakers, designed to work with SS amps, have less efficient speakers by design which dampen the boosts at resonant frequencies. Guitar speakers are efficient and the boosts are more pronounced.
So when you connect a tube amp to a guitar cabinet, you get a nice low and high boost, which sounds pleasant to your ear. When you connect an SS amp, you don't get it, and maybe even get dampened resonant frequencies, thus lack of punch.
Specifically to alleviate this "problem", there's the Speaker page which allows control over those resonant frequencies, as people have mentioned here already. They are set for some generic cabs by default which aren't the same as YOUR cab. When there's a mismatch, your Axe and your Matrix try to boost your speaker where it's not at its resonant frequency, and it just cannot react properly. So it sounds like it has no "soul". Find those frequencies, and that "soul" will return.
What I find puzzling is this. Constructing a 100% voltage driver using tubes is difficult or impossible, as far as I understand. But it's easy to make an SS based current driver. So there's nothing that is preventing manufacturers from making SS amps that do react "properly" to cabinet impedance curves. Granted, it's not really needed in hifi, but for guitar applications it is useful, and manufacturers such as Matrix could have done it. Heck, with an SS amp this can be implemented easily enough to have a switch with 2 modes.