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For tests such as chi square, the significance level is usually 0.05, but I've seen some people interpret p-values for Somers'd such as .000 as significant, displaying the results similar to this: (d= .603, p<.0005)

So is the significance level for Somers'd .0005? If so, is this the same with other similar ordinal measures of association such as gamma and Kendall? Thanks!

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The significance level does not depend on the type of test. It's arbitrary and set by the investigator.

And p = 0.000 is below both 0.05 and 0.005.

Displaying it as < 0.005 is a bit unusual but is not related to the type of test, either. You could do the same with any p value.

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