I am trying to see if there is any relationship between two values obtained by entirely different means using $\varphi=\frac{1+\sqrt5}2, \pi, e,$ etc.
In the first equation, I was finding the base of a Kepler Triangle so that the length of the sides equaled the area. That led me to the value $2\sqrt{2+\sqrt5}+2$ [approximately $6.1163420545$].
That value is the base of a right triangle; that same value multiplied by the square root of $\varphi$ is the side $b$; and the same value multiplied by $\varphi$ is the side $c$.
The second equation is simply $\pi^{e/(e-1)},$ which is approximately $6.1161695807$.
Although this could be (and probably is) completely coincidence, the figures were calculated using Microsoft Excel and are therefore very limited in their accuracy.
So given the uncertainty:
1) Is it possible that these two numbers are actually identical, which would be shown if I were to be using a program that didn't truncate values? (I find this unlikely)
2) Is there any relation between these two values that could provide a clue as to why they are so similar (or is it just complete coincidence)?
Your two numbers differ at the fourth decimal, so all they have in common is
6.116....Four digits of agreement is something we should expect to arise as a matter or random chance, unless the original expressions are picked from a pool of possible expressions with much fewer than 10,000 members.
Your expressions here look complex enough that it's easy to imagine 10,000 other "just as nice" expressions, with different numeric constants or slightly different arithmetic operations or combinations of operations.
Thus is shouldn't be a surprise to find a 4-digit coincidence.