The Solomani IMTU aren't (or weren't - I'm playing TNE era now) really the monolithic bloc of racists that the TI sees them as. It's a gross oversimplification as one might expect, nor are they really fascists, either. However, they're not as liberal and forward-thinking as they like to believe, either. They are also stuck in hidebound traditionalism nearly as bad as the Vilani in some areas.
The Solomani are a nation more because of a shared sense of cultural identity than anything else. The Solomani sphere is entirely too large for a true central government, so most political matters are handled on the world or cluster of worlds level.
IMTU, the Solomani Party was indeed voted into power. It wasn't really long ago (in Megatraveller terms) that the Solomani had many political parties and vigorous public debate. The Solomani Party began as a nationalist movement when the power of the Solomani nobles was broken in Capitol. The Solomani nobles felt that if the ingrates in the TI weren't going to listen to them, they would create their own nation around Sol. These nobles (wealthy businessowners and such) leveraged their power into the first multi-planet political party. By using their money for widespread political initiatives, they fostered a widespread belief that the TI had somehow "taken" something from the Solomani, then when they were done, they kicked the Solomani out and got in bed with the Vilani.
Then began the Solomani Rim War. The fractiousness of the Solomani political system was causing problems so the "one race, one party" line actually was developed as a method to foster greater unity in the Solomani sphere during the Rim War - political parties other than the Solomani Party at first voluntarily folded themselves into the party as a method of showing unity. However, the war ended in defeat (something the Solomani weren't ready for) and much of the sphere was occupied, and the "emergency wartime measures" became law, including things like near unlimited power for SolSec, the Solomani Party being the only legal one, etc. The shock and depression of losing the war made people accept things. But so long after the Solomani Rim War, there are many who clamber that the Solomani Party not only lost the war, they haven't done a thing to get Sol back again and think it's time for change. So far, fear of SolSec and peer pressure has kept these people quiet, but there's an increasing number of factions in the Solomani parliment that feel a return to plurality and curbs on SolSec's power is in order.
However, other political parties do exist. It is simply that to join the Solomani Parliment on Home as representative of your world or consortium, you must be a member of the Party. You can still be a member of some other political party on your homeworld, and indeed in the "factions" of the Solomani Party there are still other pan-Sphere political parties, they simply refer to themselves as "factions" and profess loyalty to the Solomani Ideal (which varies from person to person).
As for Solomani racism, it is justified by archaic wording in the Solomani Constitution. Even before the Rule of Man, new colony worlds were required to adopt a basic constitution as the basis of their government. Worlds could amend this constitution as they saw fit, but the basic blueprint was the same across all worlds because of this. The problem was that Solomani Constitution was written before there was extensive contact with non-human sentients. Though racial and sexual equality was long de facto on Terra, there was an oversight about non-humans. So the Constitution in regards to basic rights and so forth reads as "human" instead of "sophont." Thus, in the section about voting for instance, it reads "No human shall be denied the right of franchise based upon gender, religion, or race."
Much like debates about the "right to bear arms" in the US Constitution, this archaic wording allows racists to deny non-human sophonts the right to vote and basic legal rights as they are not "humans." Contrary to popular belief in the TI, most planets in the Solomani sphere have ratified amendments that define "human" as any sophont. Of course, not all worlds do this (and not necessarily because they are racist - many extend such rights as courtesy but simply loathe to change the constitution out of conservatism) and the TI likes to play these worlds up. It's also true that there are no guarantees for rights of non-humans as a whole in the Solomani Sphere. However, it can be safe to say that most Solomani believe that non-human sophonts should be given the same legal rights as "humans" (especially if they are humans or uplifted by humans, such as dolphins). The issue of voting is a little more sticky and complex, however.
SolSec is another matter, however. These guys are a nightmare of war paranoia that just won't go away. Most people think of these guys as like the Waffen SS. It's not a bad comparison (though using the SS model they're more like the Allgemeine SS and the Waffen SS), but they're not the SS IMTU, they're not even the KGB, but more like Romania's Securitate combined with Bismark's System. While the most infamous side of SolSec is that these men and women could be anywhere and anyone - most of these people aren't necessarily SolSec agents but more informers and snitches. However in the dark days after the defeat in the Rim War, the Solomani Party convinced itself it lost due to factionalism in the military and they authorized the creation of an "elite" force of party loyalists. These would be the military arm of the SolSec. These guys have their own uniformed armies (complete with actual tanks and battledress), their own air forces, their own space fleets, their own marines. They are a separate military from the actual Solomani Fleet and Solomani Army and even more galling, they have better funding than the actual Solomani military. While under the direct command of the Solomani Party (without constitutional recourse), in reality SolSec has become ever more entrenched and institutionalized, becoming a state-within-a-state. SolSec has all sorts of wiggle room, legalized war powers acts, and their bureaucrats are masters of political infighting. The only thing that has prevented SolSec from ruling the Solomani sphere de facto is that SolSec itself is riven with personal agendas/petty infighting but also that SolSec heads simply lack the vision and leadership to imagine actually ruling the Sphere.
However, it's not as bad as everyone believes with SolSec. Yes, they have bright shiny toys and they have lots of troops. But the hardcore fanatical loyalist thing is gone. They're like Jannisaries in the later day - comfortable with their power and privilege and cynical about their own political cause. SolSec are essentially bullies and convinced of their own superiority, but they're not willing to die for it. In reality, even with inferior equipment, the Solomani Army and Fleet could kick SolSec's butt. SolSec's army is mostly there to bully the common people and so on - not people who are truly trained, dedicated, and willing to accept losses. This doesn't make them any less dangerous to people like the PCs, however.