Let $E$ be an elliptic curve over a number field $F$ and $p$ a prime such that $E$ has good ordinary reduction at all places above $p$. Suppose we know that the dual $X$ of the usual Selmer group over the cyclotomic $\mathbb{Z}_p$-extension is torsion, e.g. if $F=\mathbb{Q}$.

Then we can compute the order of vanishing and the leading term of the characteristic series of $X$ as a power series in $\mathbb{Q}_p[\![T]\!]$. If the canonical $p$-adic height is non-degenerate, as expected, then the order of vanishing is equal to corank of the Selmer group over $F$, i.e. equal to the rank of $E$ if you believe the finiteness of Sha. The leading term formula, still under the assumption of the non-degeneracy of the $p$-adic height, is a $p$-adic BSD-type formula. This was first proven by Perrin-Riou and Schneider. It involves the $p$-adic regulator, Tamagawa numbers, torsion square, the order of Sha and a correction term like in the interpolation formula for the $p$-adic $L$-function at the trivial character. Of course the formula is only up to a $p$-adic unit as the characteristic series is only defined up to a unit. If there is an analytic $p$-adic $L$-function to compare to, the formula compares well with the expected $p$-adic BSD formula there.

When proving this formula, one has to compare $X^\Gamma$ and $X_\Gamma$ where $\Gamma$ is the Galois group of the $\mathbb{Z}_p$-extension. If they are finite then one can call it the $\Gamma$-Euler characteristic that one has to compute. But in the higher rank case, one uses instead the map $X^\Gamma \to X \to X_{\Gamma}$. Under the assumption that the $p$-adic height is non-degenerate, this map has finite kernel and cokernel. This is very similar to Tate's proof of the BSD formula in the function field case. Perrin-Riou generalise it to $p$-adic representations, but maybe her article "Théorie d'Iwasawa et hauteurs p-adiques (cas des variétés abéliennes)" is a good place to start.

Now to your question when the analytic rank is $1$. Let's say we are over $F=\mathbb{Q}$. Then we have a Heegner point $P$ and the rank is really $1$. One may express this formula in terms of the $p$-adic height of $P$. By the Gross-Zagier formula and its $p$-adic analogue one can compare the $p$-adic BSD formula to the usual one and hence express the leading term (which is a sort of a generalised Euler characteristic) in terms of $L'(E,1)$. I think it is best to look at the Bourbaki talk by Colmez http://www.numdam.org/item/SB_2002-2003__45__251_0/ on Kato's work where all of this is surveyed. It contains all the relevant references.