Modern Digital oscilloscopes have several fundamental blocks that process probe input data in a chain before the trace gets displayed.
Front End - analog circuitry with programmable attenuation and offsets etc. Depending on gain settings, it might have somewhat different bandwidth. And "bandwidth" is a stretchable concept, the transfer function can have gradual decline, not just "-3dB cut-off", which can be corrected later in the processing chain.
Sampling/ADC unit. In many cases the ADC samples the signal at constant (and rather high) rate, above the Nyquist frequency of anti-aliasing filter of the Front End. So the signal is usually oversampled. The rate of data storage however can be "decimated" in the process.
Data storage (memory) for raw data. Fast memories are required to store the input stream from ADC unit.
Before the data are displayed, modern scopes have the signal digitally processed. So you can correct the uneven input characteristics, put interpolations, adjust scales in all directions, and run various measuring algorithms. This is all in post-processing software, to put a nice picture on LCD.
So there are many options for software/firmware to change/extend basic scope characteristics, depending on how much you are willing to pay. Most notably the "software upgrades" are used in configuration of depth of data storage. Scopes might have the super-fast memory soldered down for the maximum already, but software enables only certain portion of it. And to get full memory, you might need to purchase a special license to use it, and it might be sold on expiration basis.
Regarding bandwidth "upgrades", if the software-based "upgrade" is offered, then the scope has a full-featured front-end that meets highest advertised parameters. However, good quality analog circuitry in high-MHz area is expensive. Manufacturers of analog-to-digital components usually have their ICs binned to different grades, and prices vary substantially. It is possible that less expensive version of a scope has the binned-down front-end components, which can be upgraded only by upgrading the hardware module.