I don't know that there's a defining physical characteristic. There are an awful lot of derailleurs that were designed for X speeds that will work just fine with X+1 or X+2. Cable pull ratio is generally the same, so if its limit screws give you enough adjustability (and many of them will), it'll work.

If I wanted to know how many speeds a derailleur was intended for, I'd look up its part number. If I want to know how many cogs it'll actually shift, I'd put it on a bike to test it.

There's no direct correlation between spindle length and front derailleur travel. The spindle length and crankset together determine the chainline, or how far the chainring is offset from the bike's centerline. A derailleur only cares about the chainrings' offset from the bike's centerline, not how much of that offset is due specifically to the spindle.

While you don't need to worry about spindle length when choosing a front derailleur, there are plenty of other things you should be aware of.

• Does the derailleur's clamp size match the diameter of the seat tube? There are at least three common sizes: 28.6 mm, 31.8 mm, and 34.9 mm. (Plus a few other possible sizes, like 28.0 mm for older French bikes.)
• What size chainring is the derailleur designed for? The curvature of the cage should be a pretty close match for the curvature of the chainring.
• Is the derailleur meant for a double or triple chainring?
• Is the derailleur "top pull" or "bottom pull"? (From which direction does the cable attach?)
• Shimano road and mountain bike front derailleurs have different cable pull ratios. Not a problem for friction shifting, but if you're using indexed shifting, you generally need to match road shifter with road derailleur or mountain with mountain.

There are at least a few more variables, too. Sheldon Brown's page titled "All About Front Derailleurs" is a great resource for more info:
All About Front Derailers