Do welding units, such as a Multimatic 220 AC/DC (PDF spec sheet), need a specific generator to power it? For example, does that unit require the Miller Bobcat 260 (PDF spec sheet)? Or can cheaper generators be used?
A welding unit needs a power source compatible with its power requirements. For the Miller Multimatic 220 AC/DC, that varies with your intended usage and amp draw. If you're intending to stick weld heavy steel at 200 A (welding current), you're going to need a 240 volt power source rated at at least 32 amps (i.e., 40 or 50). Also keep in mind your duty cycle; the welder is only good for a 15% duty cycle at that heavy power draw...10 seconds welding, 50 seconds resting.
On the other hand, if you keep your stick welding amp draw at 90A or less (40% duty cycle) or TIG welding at 130A, the welder will be perfectly happy with 120V power. (Not from a regular wall outlet, though, not at those amperages...you'd need a special 30 amp 120V outlet).
Now, while Miller very well may want to sell you their generator (and it might be worth considering to have equipment you know will work and play well together), the welder really won't care where the power comes from as long as it meets the voltage and current supply requirements. If you have another generator which is capable of supplying the load, by all means use it (Do make sure you have compatible plugs and cords first, of course!). If you're just using the machine in your own garage or shop, you really don't need a generator at all...just call an electrician to have the proper outlet installed wherever you plan to use the welding machine.
Editing To Add: I see that the Bobcat is a welding machine as well as a generator, so if you purchase it you will only need the one piece of equipment. It's up to you if you would rather have two separate machines (the Multimatic welder plus whatever lower-priced generator you choose).