Hey, a question dealing with my incredibly niche area of academic expertise!

Yes, the Coliseum did indeed have something resembling season ticket holders. To answer your question in brief, wealthy patrons could pay to have their names inscribed on specific seats, reserving them for future use. However, that was only one part of the Coliseum's ticketing system and just one way in which it resembles the major sports facilities of today.

The Coliseum had what is considered to be the first ticketing system for a sports facility. Pieces of pottery were distributed prior to events with a seat number, row, and (most importantly) an entry gate inscribed upon it. That piece of pottery acted as a ticket and granted entry through the inscribed gate and the inscribed gate only. There were 78 gates, each leading directly to their designated section. By tying entry and exit to a specific location in such a way, 50,000 people could enter within 15 minutes. The Coliseum also had private viewing boxes for VIPs and designated sections for different groups of people. Much like how soccer stadiums of today has sections specifically for supporters clubs, the Coliseum had designated areas for students, members of religious orders, and soldiers, in addition to its civilian areas.

Beyond its ticketing and seating systems, there were many aspects of attending an event at the Coliseum that would be recognizable to a modern day sports fan. You would find in-seat food service and hawkers, promotional events with giveaways and contests for attendees, and security to take care of unruly fans (all the way up to archers posted with orders to shoot if need be, similar to rooftop snipers for security at major events of today). The Circus Maximus also offered seatback chair rentals along with merchandise and concessions sales, something which is common at college football stadiums today.

Source: Fried, Gil and Kastel, Matthew. Managing Sports Facilities. Human Kinetics Publishers, 2020