I think this might work. Essentially, you need to save each text separately, instead of just having one text.

const [texts, setTexts] = useState({});

const assignTutorToChildFunction = (id_child) => {
    setText({...texts, [id_child]: "Cet enfant est déja ajouté"});
};

React.useEffect(
    () => {
        let temp_texts = {};
        children.map((element, index) => temp_texts[index] = "Ajouter cet enfant");
        setTexts(temp_texts);
    },
    []
);

You are using a state variable. Once you change it, it will cause a re-render and all of your references will be updated.

Better to try to have 2 separate variables containing texts.

const tutorUnassignedText = "Ajouter cet enfant";
const tutorAssignedText = "Cet enfant est déja ajouté";

then you need a state variable to track where to show which text.

You could try something like this:

const [tutorsAssigned, setTutorsAssigned] = useState(Array.from({length: children.length}).fill(false))

and then inside your map use:

<Inner>
  <button 
onClick={() => setTutorsAssigned(prevState => {
  const tutorsArr = [...prevState];
  tutorsArr[index] = !tutorsArr[index]
  return tutorsArr;
})>
    {tutorsAssigned[index] ? tutorAssignedText : tutorUnassignedText}
  </button>
</Inner>

Each child component must have own state:

Const ChildComponent = () => {
    const assignTutorToChildFunction = (id_child) => {
      setText("Cet enfant est déja ajouté")
    }
    const [text, setText] = useState("Ajouter cet enfant");
    return (
      <StyledTab>
        <Outer style={{ backgroundColor: "grey", color: "yellow" }}>
            <Inner>
                <th>Prénom</th>
            </Inner>
            <Inner>
                <th>Nom</th>
            </Inner>
            <Inner>
                <th>Date de naissance</th>
            </Inner>
            <Inner>
                <th> Ajout de l'enfant</th>
            </Inner>
                <Outer>
                    <Inner>

                        <td>{element.firstName} </td>
                    </Inner>
                    <Inner>
                        <td>{element.lastName} </td>
                    </Inner>
                    <Inner>
                        <td>{element.birth}</td>
                    </Inner>
                    <Inner>
                        <button onClick={() => assignTutorToChildFunction()}>{text} </button>
                    </Inner>
                </Outer>
            
          </Outer>
      </StyledTab>
  ) ;
} ;

Const ParentComponent = () =>  {
  return children.map((element, index) => (
    <ChildComponent />
  ))
 }