I'm not saying that the Texas name doesn't carry weight when it comes to other things, like television deals, conference alignment, etc.
But when it comes to how opponents play your, the energy the players and fans have when you roll into town, no Texas's name does not carry anywhere near the same weight that Kansas' name does in basketball. Not in the same universe. When KU comes to town, teams play with a different energy. Fans have "black-outs", and sell out stadiums that 95% of the time they can't get half full. The same does not happen when Texas comes to town.
Now, if Shaka can get the program to play like this consistently on a yearly basis, where Texas is going far in the tournament every year, and they're finishing at the top of the conference or competing with Kansas for the top spot on a yearly basis, then yes Texas will get that kind of response out of opposing fans and players. But right now you don't. Last year, watching Baylor play, and they've been a consistently good team, even if rarely at the top of the conference, they were still a top 5 team, and I watched a lot of games where they played on the road and the crowd was lethargic and sparse, where as when KU went to those same places, we faced sellouts and loud crowds. To garner that kind of attention, it requires not only having a good team, but it also requires tradition, and a history of consistently being at or near the top. When you've been doing it that long, you can get that kind of response from opposing players/fans even when your team isn't that great.
KU on the other hand, if we turned it around in football, we couldn't garner that kind of response from opposing fans and players. Not until we did it for a long time. Where as Texas, even though you're trying to rebuild, you're still getting that kind of response from opposing players/fans on the football field.
I watched Oklahoma play a couple of weeks ago, and they were going through the motions. Looked like they didn't want to be there. Their defense was sluggish, slow, and undisciplined. Today, they played perhaps the best defense we've seen this year. They were jumping passing lanes, getting their hands on every ball, jumping into the stands after loose balls, diving on the floor, etc. They gassed themselves with their effort, but they wanted that W. It was pure effort. KU faces that every game in basketball. And even though it's kind of annoying, I'm glad this is the case, because it's a sign that your program has been really good for a long time. If Shaka can keep recruiting well, and get the program consistent, Texas is going to face the same on the court.