No such analog filter IC exists because the topology of the different kind of filters is too varied to combine all of them into a single IC. It also seems the ICs capable of more than one kind of filters need a LOT of external components to set everything up. Analog filter design is still labourous (and seriously out of my league).
Only digital FIR filters come close to what you speak of. They requires an ADC front-end followed by a processor to do the filter calculations. The more ideal the filter is, the more processor speed and memory is required. These filters can achieve perfect group delay. Throwing more calculations into the filter (aka more processor resources), you can approach a flat passband, narrow transition band, of arbitrary order, and very accurate frequency cutoffs for low pass, highpass, bandpass, and a bunch of other types of filters I've never heard of. It's all in the software. If the hardware is powerful enough, you just swap the software for a different filter. They are slower than analog filters though since they inherently have a processing latency (which is why you need a faster processor and ADC to process data at higher rates.
I suppose it's possible to add on a DAC to output the filtered signal in analog form, but no one ever does to my knowledge. Changing the information into digital form to be processed is usually the goal.