"NodeMcu" is the name of a firmware originally for the ESP8266 microcontroller, with support for the ESP32 microcontroller added more recently, that allows you to program these microcontrollers using the Lua programming language.
In the early days of the hobbyist community's adoption of the ESP8266, ESP8266 development boards were marketed with the name NodeMcu and the term "NodeMcu" started to be associated more with the hardware than the firmware. It is common for people with no interest in the Lua programming language to buy "NodeMcu" boards and program them with the Arduino IDE, which erases the NodeMcu firmware from the board. To make things more confusing, multiple companies marketed boards with the name "NodeMcu", but with small hardware differences. Over time, the NodeMcu boards became less popular in the Arduino world because the similarly priced WeMos D1 Mini came out with the same functionality, but in a smaller form factor. Now, the "NodeMcu" name is being used to market ESP32 boards as well.
ESP32 is the name of a microcontroller made by Espressif. You will find this microcontroller on a variety of development boards, in a variety of forms. You can also purchase the ESP32 module, without the support components provided by the development boards.
ESP8266 is the name of a microcontroller made by Espressif. You will find this microcontroller on a variety of development boards, in a variety of forms. You can also purchase the ESP8266 module, without the support components provided by the development boards.
Pictures I have seen seem to show that the ESP32 and the NodeMcu are the same, yes/no?
It seems that the NodeMcu uses a Lua programming language rather than the arduino development system in C++.
The NodeMcu firmware is for use with Lua. The NodeMcu hardware can be used with Lua, C++, C, assembly, Python, and probably a few other languages.
Can you consider a ESP32 an arduiino with a wi-fi chip?
it seems that the ESP32 supports WiFi AND Bluetooth, is this correct?
I see no mention of Bluetooth on the NodeMcu.
What do you mean by "NodeMcu"?
If you're talking about the NodeMcu firmware, does it provide access to the ESP32's Bluetooth functionality? I have no clue.
If you're talking about the NodeMcu hardware, was the board you were looking at an ESP8266 board? If so, then it does not have Bluetooth functionality. Only the ESP32 has Bluetooth.