Matlab Estimated Alpha vs Tau-Alpha Formula in Copulas

Question

I am new and have been trying to understand copula. Im only using Clayton, Gumbel and Frank for now. Here's my situation.

1. I have two data x and y (log return series). I try fitting them a copula in matlab "copulafit" command. It gives me the best fit with AIC/BIC and an estimate of parameter Alpha. Let's say for now the program chooses "Gumbel" as best fit among three. So now I have Gumbel and estimated parameter Alpha.
2. Then I saw some papers using Kendall Correlation (Tau) of the x & y data then getting the Alpha using Alpha = (1-tau)^-1 for gumbel.
3. I compare the computed values of the two Alpha. Which then gives different values.

Question:

1. Are these supposed to give different values?
2. I think I read something like that the Tau in the estimate is different from the Tau on the x and y data. Is that correct?
3. Finally, now having the Gumbel and two values of Alpha, which Alpha should I input in my formula of Conditional Probability for the Gumbel formula? which is more correct value to use? How should I interpret it?

Additional Info/What I did:

1. I try using the example in matlab where it fits x and y data to t-copula. The estimated rho (Pearson linear correlation) is almost equal to the rho of the x and y data. So there is definitely Im missing since the two alpha mentioned in question is too far apart.