The implementations of the two Ticker.h files are a bit different. On ESP8266, the once method expects type "callback_function_t" where callback_function_t is defined as:

typedef std::function<void(void)> callback_function_t;

On ESP32, it expects type "callback_t" which is defined as:

typedef void (*callback_t)(void);

In your code example, std::bind provides the std::function type that is expected. This is not the case for the ESP32 Ticker.h, it expects a function pointer. You have two options:

1. Instead of having the onTickerCallback function part of the Test class, just create a free function for the callback. (Note that this is only acceptable if the callback doesn't need to be part of the Test class).

#include <Arduino.h>
#include <Test.h>
#include <functional>

// for ESP8266: https://github.com/esp8266/Arduino/blob/master/libraries/Ticker/src/Ticker.h
// for ESP32:   https://github.com/espressif/arduino-esp32/blob/master/libraries/Ticker/src/Ticker.h
#include <Ticker.h>

Ticker ticker;

void callbackFunction() {
  Serial.println("Hello");
}

void Test::start(){
  ticker.once(5, callbackFunction);
}

2. Create a free function that takes an instance of test and use this to call your function. (Note this will also require onTickerCallback to be public, no real way around this that I can think of).

#include <Arduino.h>
#include <Test.h>
#include <functional>

// for ESP8266: https://github.com/esp8266/Arduino/blob/master/libraries/Ticker/src/Ticker.h
// for ESP32:   https://github.com/espressif/arduino-esp32/blob/master/libraries/Ticker/src/Ticker.h
#include <Ticker.h>

Ticker ticker;

void callbackFunc(Test* testInstance) {
  testInstance->onTickerCallback();
}

void Test::start(){
  ticker.once(5, callbackFunc, this);
}

void Test::onTickerCallback() {
  doSomething();
}

void Test::doSomething() {
  Serial.println("Hello");
}

Bonus: After thinking about it for a second, you could use a lambda instead of creating the function (note you need to have a + sign before the lambda in order for it to work as a function pointer). This would look like:

#include <Arduino.h>
#include <Test.h>
#include <functional>

// for ESP8266: https://github.com/esp8266/Arduino/blob/master/libraries/Ticker/src/Ticker.h
// for ESP32:   https://github.com/espressif/arduino-esp32/blob/master/libraries/Ticker/src/Ticker.h
#include <Ticker.h>

Ticker ticker;

void Test::start(){
  ticker.once(5, + { testInstance->onTickerCallback(); }, this);
}

void Test::onTickerCallback() {
  doSomething();
}

void Test::doSomething() {
  Serial.println("Hello");
}

ESP32 ticker attach functions take pure function pointers with at max one argument type -- not a std::function/ std::_Bind_helper that std::bind produces.

See code and types at https://github.com/espressif/arduino-esp32/blob/master/libraries/Ticker/src/Ticker.h#L38.

For the ESP8266 however, they do use a std::function:

https://github.com/esp8266/Arduino/blob/master/libraries/Ticker/src/Ticker.h#L36

And that's why it works on ESP8266 and not on ESP32 -- either adapt the Ticker library or rework your own code to not use member functions of classes.

Edit: You can easily rewrite your code to use a static helper function with one argument (the watchdog object itself) which then just calls the member function of that object. For this to work, the called function must be public. So a rewrite like

static void watchdog_timer_triggered_helper(myIOT32* watchdog) {
   //needs to be public!
   watchdog->_feedTheDog();
}

void myIOT32::_feedTheDog()
{
  _wdtResetCounter++;
  if (_wdtResetCounter >= _wdtMaxRetries)
  {
    sendReset("Dog goes woof");
  }
}
void myIOT32::_startWDT()
{
  wdt.attach(1, &watchdog_timer_triggered_helper, this); // Start WatchDog
}

void myIOT32::sendReset(char *header)
{
  char temp[150];

  sprintf(temp, "[%s] - Reset sent", header);

  if (useSerial)
  {
    Serial.println(temp);
  }
  if (strcmp(header, "null") != 0)
  {
    pub_msg(temp);
  }
  delay(1000);
  ESP.restart();
}

Since both the ESP32 and ESP8266 versions implement the

    template<typename TArg>
    void attach(float seconds, void (*callback)(TArg), TArg arg)

templated function this should work on both platforms now.