First of all, Marius' consecutive consulships were irregular, but absolutely legal. Roman laws during the Republic were more like guidelines, really. Well, it's not that simple. To be illegal, an act must be in conflict with a law. The way to determine that was by having a trial. But Rome had no public prosecutors, so for there to be a prosecution you had to have pissed someone off enough to take you to court. Sulla, after his dictatorship, proscriptions, and reorganization of the state, appeared in the forum and presented himself for prosecution, and no one prosecuted. Sulla's actions were nasty and brutal, and his reforms were repealed by his own allies, but no one talks about him as doing illegal things.

The second point is with reference to the consul and other officers with imperium specifically. The whole set up of the Centuriate Assembly is military - people vote in a military space, the Campus Martius, outside the pomerium, the boundary of civic space, they vote in their military organization of centuries, and they elect magistrates with imperium - the quality necessary for military command. This is the army appointing its own officers. That in fact is Livy's justification for the wealth classes (whether that's how they originated is another question). The wealthy contribute more to the army and so have a greater say in the election of magistrates that lead them.

The principle of the army selecting its own commanders in contravention of the law, tradition, whatever, is well demonstrated in the case of Scipio Aemilianus, who was elected to command Punic 3 despite not being old enough to meet the age requirement.

Finally, a LOT of consuls held second or third consulships. Here's a list; the Roman numeral after the name indicates which time the person was holding the position. Some of these are straight up legendary (I doubt Publius existed, for instance), but L. Papirius Cursor was consul for the 5th time in in 313 and I don't see a reason to doubt that little bit of information.

So, Marius' consecutive consulships were irregular, both in being consecutive and so many, and being elected in abstentia, but rules restricting who can stand for consul had been bent for ages in the right circumstances. He was never tried. His tenure as consul for six consecutive years, and seven in total is only without precedent in how far they bent the rules, not in the bending itself.

EDIT: Oh, I noticed another misconception. The Senate didn't elect consuls. The people in their capacity as the army in the Centuriate Assembly elected consuls. It was traditional to have the approval of the Senate, and the officiating magistrate generally had to accept someone as a candidate, but the Senate didn't have any legal authority to bar someone from standing for consul, and the officiating magistrate could be compelled by popular opinion (as happened with Scipio Aemilianus) to accept someone they would rather reject.