No, it is not the same thing.
This question has gone unanswered so I dug into it a bit,
Based on the below excerpt from the ESRI website coupled with an excerpt from Fotheringham. It seems to me that the effective number is the effective number of parameters, per explicit mention in Fotheringham et al. 2002 (p. 92) as well as the fact that ESRI worked or contracted with Fotheringham and friends to develop the built-in tools that come with ArcGIS.
The effective degrees of freedom would then be
n - effective number of parameters. Where
n is the population or number of datapoints.
See the below excerpt from the diagnostic output of the GWModel GWR package in R from my own GWR:
Number of data points: 62
Effective number of parameters (2trace(S) - trace(S'S)): 39.07396
Effective degrees of freedom (n-2trace(S) + trace(S'S)): 22.92604
That being said, I do not know to what extent one may or may not calculate p-values or determine cut-offs for other statistical tests using the effective degrees of freedom, but I believe this answers the question of whether they are the same.
"The effective number of parameters in a GWR is often not an integer but varies between k (when the bandwidth tends to infinity) and n (when the bandwidth tends to zero)."
(Fotheringham et al. 2002, p 92)
"EffectiveNumber: this value reflects a tradoff between the variance
of the fitted values and the bias in the coefficient estimates, and is
related to the choice of bandwidth. As the bandwidth approaches
infinity, the geographical weights for every observation approach 1,
and the coefficient estimates will be very close to those for a global
OLS model. For very large bandwidths, the effective number of
coefficients approaches the actual number; local coefficient estimates
will have a small variance but will be quite biased. Conversely, as
the bandwidth approaches zero, the geographical weights for every
observation approach zero with the exception of the regression point
itself. For extremely small bandwidths, the effective number of
coefficients is the number observations, and the local coefficient
estimates will have a large variance but low bias. The effective
number is used to compute a number of diagnostic measures."