Please try this code:

import axios from "axios";
import {AxiosError} from 'axios';

    axios
        .post("http://localhost:5000/run", payload)
        .then((response) => setOutput(response.data.output))
        .catch((error) => {
            if (!error?.response) {
                setOutput("No Server Response");
            } else if (error?.code === AxiosError.ERR_NETWORK) {
                setOutput("Network Error");
            } else if (error.response?.status === 404) {
                setOutput("404 - Not Found");
            } else if (error?.code) {
                setOutput("Code: " + error.code);
            } else {
                setOutput("Unknown Error");
            }
        });

You can use axios.isAxiosError(error) to make sure that the error is type of AxiosError.

Axios error types:

static readonly ERR_FR_TOO_MANY_REDIRECTS = "ERR_FR_TOO_MANY_REDIRECTS";
static readonly ERR_BAD_OPTION_VALUE = "ERR_BAD_OPTION_VALUE";
static readonly ERR_BAD_OPTION = "ERR_BAD_OPTION";
static readonly ERR_NETWORK = "ERR_NETWORK";
static readonly ERR_DEPRECATED = "ERR_DEPRECATED";
static readonly ERR_BAD_RESPONSE = "ERR_BAD_RESPONSE";
static readonly ERR_BAD_REQUEST = "ERR_BAD_REQUEST";
static readonly ERR_CANCELED = "ERR_CANCELED";
static readonly ECONNABORTED = "ECONNABORTED";
static readonly ETIMEDOUT = "ETIMEDOUT";

You need to try using response.response.data.error for displaying error in your case instead of response.data.err.stderr. So here you are getting error data from response :

try{
    const {data} = await axios.post("http://localhost:5000/run",payload);
    setOutput(data.output);
}
catch ({ response }) {
  if (response) {
    const errMsg = response.response.data.error;
    setOutput(errMsg);
  } else {
    setOutput("Please retry submitting.");
  }
}

Reference discussion here.

Here is the solution that worked . Thanks all for your inputs

try{
    const {data} = await axios.post("http://localhost:5000/run",payload);
    setOutput(data.output);
    }
    catch (error) {
      if (error.response.data) {
        setOutput(error.response.data.err.stderr);
      } 
      else {
        setOutput("Error connecting to server!");
      }
    }