They're similar but not that similar. I currently live in Columbus and have been to Indianapolis a few times so I feel like I'm a good person to answer the question. They both look similar in urban form from aerials and I've confused the two on a map before when I wasn't paying close attention. They're culturally similar although Columbus feels more diverse and liberal. Indianapolis feels more conservative but weirdly more progressive.
Both cities annexed huge portions of the metro area. I think Indianapolis is consolidated with Marion County, but Columbus is in three different counties and has a spider web of a corporate boundary with lots of enclaves like Upper Arlington and Bexley and random townships everywhere.
They both are not very rust belt but they both still have really awful neighborhoods that experienced decline and abandonment but they hide it well. Columbus more so than Indy. Despite both having a reputation for being new tech hubs and being more white collar than their neighbors like Pittsburgh and Cleveland, they still have a lot of Detroit looking neighborhoods if you know where to look.
The wealthier parts of both cities are to the north. Carmel, Indiana is fairly similar to Dublin, Ohio. The undesirable portions of both cities are the southern portion of the metro area. The Columbus area is lucky to have Grandview Heights, Upper Arlington, and Bexley as separate cities so close to the urban core, having those wealthy well run cities so central with separate leadership gives the inner city a nice mix of incomes and intact homes. If Columbus had consolidated with Franklin County I'd imagine those residents would have jumped ship for the Delaware County suburbs. Indianapolis doesn't seem to have a higher income neighborhood close to the downtown core.
Indianapolis has a much nicer downtown. The cultural trail, the new infill, the overall cohesiveness and interconnectedness with other urban parts of the city is really good. They seem to care more about public investments in the downtown in Indianapolis and they've put a lot of work into a new zoning code and comprehensive plan. Columbus has an outdated code and no political will to improve things to get better development or make big investments on the scale of the bus rapid transit Indy got or their separated bike lanes. Columbus is seeing a lot of investment in the downtown area but there are huge swaths of surface lots and it doesn't feel like a lively desirable place. After 5 it empties out very quickly.
Indianapolis doesn't have a Short North / High Street corridor equivalent but Columbus would be a lot more impressive if the downtown had seen similar levels of investment to Indianapolis. Most of the neighborhoods in Indianapolis are unremarkable. I am not excited by the types of homes in Indianapolis at all. One of the biggest assets of Columbus that sets it apart from Indy is the neighborhoods. There is nothing like German Village or Victorian Village in Indianapolis. Clintonville looks like a midwestern SE Portland. Columbus overall feels like an older more established metro area. Suburbs like Worthington feel like Connecticut, there are random streetcar suburban retail strips in Columbus with very old buildings. You don't really see this in Indianapolis on the scale of Columbus.
Some pictures by me so you all can compare, contrast, discuss:
German Village - Columbus
Downtown - Columbus
Short North - Columbus
Clintonville - Columbus
Downtown - Columbus
Fountain Square - Indianapolis
Downtown - Indianapolis
Lockerbie Square - Indianapolis
Downtown - Indianapolis
Fountain Square - Indianapolis
Downtown - Indianapolis