Do all microcontrollers from one family have the same processor core?
AVR is a bit of a weird example because it's a single-vendor CPU core -- it's only present in certain parts made by Microchip (formerly Atmel). That being said, I believe you are correct that the parts you mentioned all have very similar processor cores; the differences between those parts are primarily in the peripherals and memory size.

A better example of the distinction between a microcontroller family and a processor core is in ARM microcontrollers. There are many families of ARM microcontrollers (ST STM32, Atmel SAMD, NXP LPC, and many others -- some of which are divided into many subfamilies as well) but a much smaller number of ARM processor cores (Cortex-M0, M4, M7, etc), all of which are licensed from Softbank (formerly ARM).


  Do AVR 8 bit and AVR 32 bit families have different processors?


Yes. AVR8 and AVR32 are almost completely unrelated, beyond that they were both designed by Atmel.