How to get a user inputed string variable to be the name of a new object in a class

Variable names don't really exist while your code is running: they are used by the compiler to create storage locations which may be in memory, but may be entirely within the processor which has a large set of registers that have nothing to do with memory. The actual names you give them are fixed when you compile your app, and are discarded when the EXE file you run is created.

You can't "create a variable" while your code is running, you create a storage location (on the heap for example) and store the location of that in a variable or a collection (a reference to which which is stored in a variable).

Think of variables like books: the book has a title (the variable has a name), and you can have many books on a shelf (many variables in an app), but to add information to it you don't magically create a new book, you open an existing one and write on a page. Books are variables in your code, pages are the storage locations within the book - and just as you add info by finding a blank page to write on, in your code you add a new storage location to a list (for example) and store the info in that.

The class contains members which can include the name, address, age, destination, and so on - so you create an instance of the class and fill in it's individual info for each user you want to add to your app!

As Griff stated, you cannot do that.
An alternative could be using an unordered_map to link the user supplied string to newly constructed object. Try

#include <iostream>
#include <unordered_map>
using namespace std;


class Airplane
{
  unsigned tons;
public:
  Airplane(unsigned tons):tons(tons){}
  Airplane():Airplane{100}{}
  unsigned get_tons() const {return tons;}
};


int main()
{
  string ap_name;
  unsigned ap_tons;
  unordered_map<string, Airplane> m_ap;

  m_ap.emplace( "plane1", 100 );
  m_ap.emplace( "plane2", 120 );
  m_ap.emplace( "plane3", 130 );

  // example of look-up by name
  cout << "weight of plane2 is " << m_ap["plane2"].get_tons() << "\n";


  // user will name next airplane
  cout << "name your airplane: ";
  cin >> ap_name;
  cout << "specify its weight (tons): ";
  cin >> ap_tons;

  m_ap.emplace( ap_name, ap_tons );


  for (auto const & [name, plane] : m_ap )
    cout << "m_ap[" << name << "] weights " << plane.get_tons() << "\n";
}