Consider this code (check out the comments):

import React, { useState } from "react";
import Select from "react-select";
import "./styles.css";

export default function App() {  
  const [value, setValue] = useState();
  const options = [
    {
      label: "first option",
      value: 1
    },
    {
      label: "second option",
      value: 2
    },
    {
      label: "third option",
      value: 3
    }
  ];

  // destructuring the object to get 'value' property
  // (the passed object looks like { label, value } )
  const onSelect = ({value}) => {
    // here the 'value' variable is being set
    console.debug("selected value:", value )
    setValue(value);
  };

  // destructuring event object to get 'target' property
  const onBlurValue = ({target}) => {
    console.log("onBlur target value:", target.value);
    // the value is not updated yet, so it holds the previously set value
    console.log("onBlur value:", value || "undefined");
  };

  return (
    <div>
      Current value is: {value || "undefined" }
      <Select
        // the value prop does nothing here
        // value={options.label}
        options={options}
        onChange={onSelect}
        blurInputOnSelect
        onBlur={onBlurValue}
      />
    </div>
  );
}

As @a.mola said - the setValue hook is asynchronous, and the value variable isn't updated yet when the onBlur event fires, so it holds the previously set value.

I'm not sure about what are you trying to achieve within the onBlur event, it may not be the right place to do it.

If you need to output the value somehow - you can do so within the return part (as is done in code above).

If you need to validate the value or do something based on its newly selected value - you can do so within the onSelect function.

You can add an instance variable to store the selected value (maybe not the best solution but it should work)

export default function App() {
  const [value, setValue] = useState();
  const valueRef = useRef();  

  const onSelect = value => {
    valueRef.current = value;
    setValue(value);
  };

  const onBlurValue = () => {
    console.log(valueRef.current);
  };

  return (
    <div>
      <Select
        value={value}
        options={options}
        onChange={onSelect}
        blurInputOnSelect
        onBlur={onBlurValue}
      />
    </div>
  );
}

The problem with this solution is that you store the selected value in two places (the state variable and the reference instance variable).

The question is, why do you need the get the selected value inside the onBlur handler ? The onChange handler might be sufficient in your case, isn't it ?

That might be a problem with React of that version, or conflicting between react-select and that specific version of react

Try to upgrade both react and react-dom to the latest version (17.0.2) and react-select to 4.3.1 solve the issue.

Working Example: