React and Chartjs not updating from dynamic data through api

You can attach the map referance to the class and once data changes use this referance. Modified your code.

class CountByUserGraph extends React.Component {

  this.massPopChart;

  constructor() {
    super();
    this.state = { selectedOption: "A" };
  }

  componentDidMount() {
    this.makeApiCall();
  }

  makeApiCall = () => {
    countByUserGraph(this.state.selectedOption) // API CALL
      .then(response => {
        // recieves data
        this.getChartData(data);
      })
      .catch(err => console.log(err));
  };
  getChartData = data => {
    let myChart = document.getElementById("myChart0").getContext("2d");
    const CountValue = data.map(a => a.countValue);
    const CountMonth = data.map(a => a.month);
    var data = {
      labels: CountMonth,
      datasets: [
        {
          label: "Count",
          data: CountValue
        }
      ]
    };

    var option = {
      responsive: true
      // Options
    };

    if(this.massPopChart){
       massPopChart.data = data;
       //massPopChart.config.data = data; //use this if above code does not work
       massPopChart.update();
    }
    else{
       this.massPopChart = new Chart.Line(myChart, { data: data, options: option });
    } 
  };

  handleOptionChange = changeEvent => {
    this.setState({ selectedOption: changeEvent.target.value }, function() {
      this.makeApiCall();
    });
  };

  render() {
    return (
      <div>
        <form className="graph-filter">
          <div>
            <input
              type="radio"
              id="A"
              name="A"
              checked={this.state.selectedOption === "A"}
              onClick={this.handleOptionChange}
              value="A"
            />
            <label htmlFor="A">A</label>
          </div>

          <div>
            <input
              type="radio"
              id="B"
              name="B"
              checked={this.state.selectedOption === "B"}
              onClick={this.handleOptionChange}
              value="B"
            />
            <label htmlFor="B">B</label>
          </div>
        </form>
        <canvas id="myChart0"></canvas>
      </div>
    );
  }
}

you can call the update function, see the docs here

so in your codesnippet you can update the chart as shown below

let massPopChart = new Chart.Line(myChart, { data: data, options: option });
// for example 
massPopChart.options.title.text = 'new title';
massPopChart.update()

similar way you can update the data also.