I am not sure but try this

import axios,{ AxiosRequestConfig } from 'axios';

const config:AxiosRequestConfig = {
        method: "delete",
        url: url,
        headers: { 
            "Authorization": `Bearer ${elasticPrivateKey}`, 
            "Content-Type": "application/json",
        },
        data: data,
    };

You can extend any library types, by using the typescript decleration merging feature. (docs)

So this should do the job:

// theFileYouDeclaredTheCustomConfigIn.ts
import 'axios';
declare module 'axios' {
  export interface AxiosRequestConfig {
    handlerEnabled: boolean;
  }
}

Go to the type definition of AxiosRequestConfig, you can see that, everything, including axios headers is optional:

interface AxiosRequestConfig<D = any> {
  url?: string;
  method?: Method | string;
  baseURL?: string;
  timeout?: number;
  timeoutErrorMessage?: string;
  withCredentials?: boolean;
  // ... omited 
}

When creating an axiosInstance and adding config, the config should be returned for the instance to pick up the configuration:

const axiosInstance = axios.create();

axiosInstance.interceptors.request.use((config) => {
  if (config.headers) { // <- adding a check 
    config.headers["x-projectkey"] = 1234; // no errors
  }
  config.withCredentials = true; // to include credential, it should be outside of "if" block 
  return config;
});

Then in every request you made with the axiosInstance, it will include the x-projectkey header and credentials:

axiosInstance.get("url");

There is an ability to set different types to input an output data (axios definitions):

post<T = never, R = AxiosResponse<T>>(url: string, data?: T, config?: AxiosRequestConfig<T>): Promise<R>;

So, in your case it would be something like:

export interface UserRegistrationModel {
  email: string;
  password: string;
  firstname: string;
  lastname: string;
}

export const register = async (user: UserRegistrationModel) => {
  const { data } = await http.post<UserRegistrationModel, AxiosResponse<{ accessToken: string }>>("/users", user);
  return data;
};