Without belaboring this too much, there is a very important distinction that you are forgetting/omitting here. Your examples of OU and Florida are not really pertinent to this discussion because they are a) football schools first and foremost as you mention and b) are able to be something other than just a 'basketball school'. Our friends in Ames do not have that same luxury.
Very simply stated, isu is thought to be a 'basketball school' as much out of necessity as anything. They certainly can make no claim to be a football school and on the men's side at least, they have declined precipitously in wrestling so that leaves basketball. The questions and comments I posed earlier point to how elastic some of the isu faithful can be in applying that term to different teams. The recent success isu has had on the hardwood is impressive, but not exceptional by any means. Certain isu fans want to point out that Iowa has only won one NCAA game in the past however many years... that is still one more win in the tourney than isu had in March of this year. Iowa has as good or even better history in the tourney than does isu, but I do not hear a lot of Hawkeye fans proclaiming Iowa to be a 'basketball school'.
In general, the administration at isu worries excessively about what anyone else thinks about them. They would be much better served to just go out and focus on what they are able to accomplish.