Why would any other advanced civilization somewhere in the observable universe have the same numerical value for the fine structure constant?

Yes, they would. The point of physical constants is that, unless the laws of physics are variable, then physical constants are the same anywhere in the universe, so any other civilisation should have the same values for them as we do. It doesn't matter how advanced you are, what you are measuring is the same.

So for instance, light travels at the same speed no matter where you are, so any alien civilisation should have the same value as we do.

They probably don't use the same units as we do- so they wouldn't say "it's 3*10^8 meters per second", because they probably don't use meters or seconds. But if you converted our speed into their units, it should be the same.

Yes, of course. Just like they would have the same value for any other "constant". The speed of light, pi, absolute zero, etc. They would probably have a different measuring system than we use. But that's irrelevant. We have different measuring systems here on Earth but they all mean the same thing. 32 degrees Fahrenheit is the exact same thing as 0 degrees Celsius, even though the measuring system is different.

'cause if they didn't, it wouldn't be a constant.

Doesn't matter which system of units you'd use, it's a dimensionless property - so, whether you're here on Earth or on Sirius B - the relationship is the same.

probably for the same reason they'd have he same value for pi. Assuming the physical laws observed here obtain everywhere else, which they probably do