I don't think the Church's Thesis says they are "structuraly identical". Even basicaly. It says they are identical in power. That's a totaly different thing!
Only the procedural computation model is based on "adressable blocks of storage", on mutable variables. In the functional model you do NOT assign values to variables only to change them later, there's actually no notion of "later" in that model. You define "variables" in the mathematical sense, you define the functions as relations between the parameters and the result, but there's NO MEMORY to address.
The reason why it's possible to "speak Perl with Lisp" is that Lisp is not purely functional. It allows you to leave the functional computation model! And then you can program the procedural way.
I am NOT speaking about the difference between individual languages. I'm talking about the difference between computation models! And even though Perl does have some functional features it's basicaly procedural. And therefore in the lower levels equivalent to C or Pascal.
Object Orientation doesn't actually fit into this discussion, it's just a syntactic sugar that alows you to tie the behaviour to the data more tightly, as far as the underlying model is concerned it's still procedural. Still based on "addressable memory blocks that are being modified by a sentence of statements".
P.S.: I've been taught CS in Czech so if some of the terms look strange it's probably incorrect translation. Plus I did not have time to keep in touch with theoretical CS as I'd like to.