I checked with some console logging and noticed that the name property is not an attribute of the event.target. The event.target you have logged displays

<div class="MuiSelect-root MuiSelect-select MuiSelect-selectMenu MuiInputBase-input MuiInput-input" tabindex="0" role="button" aria-haspopup="listbox" aria-labelledby="standard-select-currency-label standard-select-currency" id="standard-select-currency">$</div>

which has attributes class, tabindex, role, aria-haspopup, aria-labelledby, and id

but the sibling of your target which is accessed by event.target.nextSibling is

<input name="mySelectBox" aria-hidden="true" tabindex="-1" class="MuiSelect-nativeInput" aria-describedby="standard-select-currency-helper-text" value="USD"></input>

or

<input name="mySelectBox" aria-hidden="true" tabindex="-1" class="MuiSelect-nativeInput" aria-describedby="standard-select-currency-helper-text" value="EUR"></input>

depending on which currency you select has attributes name,aria-hidden,tabindex,class,aria-describedby and value

then you can access the name attribute inside the handleFocus as event.target.nextSibling.getAttribute('name') or event.target.nextSibling.name

codesandbox demo

The Texfield does not directly pass the onFocus to the input. From MUI,

The ref is forwarded to the root element.

Any other props supplied will be provided to the root element (FormControl).

The onFocus is passed to the FormControl element, which is a div, not an input, and doesn't have a name. You can use the NativeSelect or pass the onFocus property to the input props, which didn't play out very well when I tried in the sandbox you provided.

The quickest solution for this is to pass a ref to the input, and on focus, get the name from the reference. This is what your code will look like then.

export default function MultilineTextFields() {
  const [currency, setCurrency] = React.useState("EUR");
  const inputRef = React.useRef()

  const handleChange = (event) => {
    setCurrency(event.target.value);
    console.log(
      "event.target ==>",
      event.target,
      "event.target.name ==>",
      event.target.name
    );
  };

  const handleFocus = (event) => {
    console.log(
      "event.target ==>",
      inputRef,
      "event.target.name ==>",
      inputRef.current.node.name
    );
  };

  return (
    <form noValidate autoComplete="off">
      <TextField
      inputRef= {inputRef}
        id="standard-select-currency"
        name="mySelectBox"
        select
        label="Select"
        value={currency}
        onChange={handleChange}
        onFocus={handleFocus}
        helperText="Please select your currency"
      >
        {currencies.map((option) => (
          <MenuItem key={option.value} value={option.value}>
            {option.label}
          </MenuItem>
        ))}
      </TextField>
    </form>
  );
}

Here is the sandbox I used https://codesandbox.io/s/material-demo-forked-zjlz3?file=/demo.js

You can pass a SelectProps to the TextField and use <option> tag instead of <MenuItem> component as I am saying below:

<TextField
        id="standard-select-currency"
        name="mySelectBox"
        select
        label="Select"
        value={currency}
        onChange={handleChange}
        onFocus={handleFocus}
        helperText="Please select your currency"
        SelectProps={{ native: true }} // ************
      >
        {currencies.map((option) => (
          <option key={option.value} value={option.value}>
            {option.label}
          </option>
        ))}
</TextField>

I modified your sandbox.