This is a great question.

Consulship in the late Roman empire had little real authority. Beginning in the Principate it became routine for the emperor to be awarded the Consulship roughly every other year as the office became defunct, and by the late empire it was essentially an honorific granted by the emperor or higher bureaucracy to those of great prestige. After the division of the empire, it became routine to select one person from the court of Constantinople and one from the court of Ravenna for the consulship, with both emperors holding the post every few years. Being consul carried with it status and carrying four consulships was still considered extremely prestigious, but the titles of actual power had transferred to the proper bureaucracy like the Magister Officiorum or particularly the post of comes et Magister Utriusque Militiae in the western administration.

The Consuls still, technically, presided over the Senate and we do see Flavius Aetius do so in 438, when the Theodosian Code was adopted, which we know because the Senate had to chant an honorific multiple times as mandated by the Codex Theodosianus and recorded in the minutes of the Roman senate. It went like this:

"Aeti aveas! Ter consulem te! Excubiis tuis salvi et securi sumus! Excubiis tuis, laboribus tuis!"

Trans: "All hail Aetius! A third consulship for you! Through your exploits we are safe and secure! Through your exploits, through your labors!" (Each line was chanted between 13 and 18 times).

The position of Consul was still extremely prestigious, it just had no function. But the Roman world built itself on the idea that prominence and fame was tied to your office and service to the empire, rather than on land ownership or fiscal holdings (which were important obviously, but not ingrained into the Roman consciousness). Eventually though it held even less meaning than it did under Aetius (or Belisarius), and by the "middle Byzantine" period the Senate ceased to exist entirely, with even its ability to propose legislature to the emperor relegated into the dustbin.

I hope this answers your question, if you want to know more about this I strongly recommend Christopher Kelly's Ruling the Later Roman Empire.