What you're probably experiencing is the result of two things: differing policies between brands (although Avis and Budget do share ownership) and differing operational models.

Avis operates on a highly integrated national system. Most of their larger locations are corporately owned. Their other locations that are franchised are not true independently-operated franchises. Instead, they are what Avis terms "agencies." Avis "agencies" run the national corporate computer systems, use the national corporate fleet, and basically outsource everything except real estate (the rental office) and hiring (sales agents and car detailers) to Avis corporate.

This allows them to do things like make a policy that allows anyone to show up at any other Avis desk and add themselves on as an additional driver, since any Avis (at least in a given region) can access information about other Avises.

On the other hand, Budget has many independent franchises around the country--independent operations that run their businesses completely separate from Budget corporate. They own their own cars, provide their own computer system, and generally do not interact with Budget corporate except to receive reservations and pay Budget corporate any franchise fees.

Although Budget in Tampa and Budget in Orlando may well both be corporate (although Tampa could be franchised, since it's not quite as "major" a destination as Orlando), this intermix of franchises around the country (that do not have access to rental data from other Budget locations) probably leads Budget to simply not allow the same thing that Avis does.

So, if Budget says no, you can't do it--don't be surprised. In fact, I'd be surprised if they do allow it. If so, count yourself lucky--but I wouldn't make definitive plans on it being possible.