Finally, after some experiments, I found, that the DAC driver functions cannot work if they are inside class.

And the function dac_continuous_write_cyclically() must be called after the channels activation:

ESP_ERROR_CHECK(dac_continuous_new_channels(&cont_cfg_0, &handle_0));
ESP_ERROR_CHECK(dac_continuous_enable(handle_0));

// After
ESP_ERROR_CHECK(dac_continuous_new_channels(&cont_cfg_0, &handle_0));

The fully working code:

The main.cpp file:

#include <Arduino.h>

#include "DAC_Continuous_DMA.h"

void setup()
{
  Serial.begin(921600);

  dac_continuous_dma_config(1000);
}

void loop()
{
  // Just for testing
  Serial.printf("Free heap size: %d\n", esp_get_free_heap_size());
  Serial.printf("%s(), core: %d: Running time [s]: %d\n", __func__, xPortGetCoreID(), millis());

  vTaskDelay(pdMS_TO_TICKS(1000));
}

The DAC_Continuous_DMA.h file:

#pragma once
#ifndef DAC_Continuous_DMA_h
#define DAC_Continuous_DMA_h

#include "Arduino.h"
#include <math.h>
#include "freertos/FreeRTOS.h"
#include "freertos/task.h"
#include "soc/dac_channel.h"
#include "driver/dac_continuous.h"

#include "soc/sens_reg.h"

#define EXAMPLE_ARRAY_LEN 512     // Length of wave array
#define EXAMPLE_DAC_AMPLITUDE 255 // Amplitude of DAC voltage. If it's more than 256 will causes

#define CONST_2_PI 6.2832 // 2 * PI

_Static_assert(EXAMPLE_DAC_AMPLITUDE < 256, "The DAC accuracy is 8 bit-width, doesn't support the amplitude beyond 255");

dac_continuous_config_t cont_cfg;

uint8_t signal_wave[EXAMPLE_ARRAY_LEN]; // Used to store sine wave values
uint8_t amplitude = EXAMPLE_DAC_AMPLITUDE;

dac_continuous_handle_t cont_handle = NULL;

void generate_waves(void)
{
    for (int i = 0; i < EXAMPLE_ARRAY_LEN; i++)
    {
        signal_wave[i] = (uint8_t)(amplitude / 2 * (1 + sin(2 * M_PI * i / EXAMPLE_ARRAY_LEN)));
    }
}

void dac_continuous_dma_config(uint32_t frequency_Hz = 1000)
{
    generate_waves();

    cont_cfg = {
        .chan_mask = DAC_CHANNEL_MASK_ALL,
        .desc_num = 2,
        .buf_size = 2048,
        .freq_hz = EXAMPLE_ARRAY_LEN * frequency_Hz / 2,
        .offset = 0,
        .clk_src = DAC_DIGI_CLK_SRC_DEFAULT, // If the frequency is out of range, try 'DAC_DIGI_CLK_SRC_APLL'
        .chan_mode = DAC_CHANNEL_MODE_ALTER,
    };
    /* Assume the data in buffer is 'A B C D E F'
     * DAC_CHANNEL_MODE_SIMUL:
     *      - channel 0: A B C D E F
     *      - channel 1: A B C D E F
     * DAC_CHANNEL_MODE_ALTER:
     *      - channel 0: A C E
     *      - channel 1: B D F
     */

    ESP_ERROR_CHECK(dac_continuous_new_channels(&cont_cfg, &cont_handle));
    ESP_ERROR_CHECK(dac_continuous_enable(cont_handle));
    ESP_ERROR_CHECK(dac_continuous_write_cyclically(cont_handle, (uint8_t *)signal_wave, EXAMPLE_ARRAY_LEN, NULL));

}

#endif // DAC_Continuous_DMA_h

Despite the I get signals on the DAC outputs (the current question is answered), the main my goal, to generate two signals on the DAC channels of ESP32, the sine and cosine ones, thus shifted by 90 deg, is not achieved.

But it is another question. And if somebody interested in this and can help me, I created new question here.