How can I search item from Searchable Dropdown per Name instead of per Id?

You have to overwrite your class's toString() so it can search both id and name
because SearchableDropdown's source code use item.value.toString()
when you pass class name it will become I/flutter ( 7352): Instance of 'VagasDisponivei'
you can copy paste run full code below

code snippet

class VagasDisponivei {
  String v_n;
  String v_id;

  VagasDisponivei({this.v_n, this.v_id});

  @override
  String toString() {
    return '${v_n} ${v_id}';
  }
}

SearchableDropdown(
              items: _vagasDisponiveis.map((item) {
                return new DropdownMenuItem<VagasDisponivei>(
                    child: Text(item.v_n), value: item);
              }).toList(),
              isExpanded: true,
              value: selectedValue,
              isCaseSensitiveSearch: true,
              searchHint: new Text(
                'Select ',
                style: new TextStyle(fontSize: 20),
              ),
              onChanged: (value) {
                setState(() {
                  selectedValue = value;
                  print(selectedValue);
                });
              },
            ),

working demo

full code

import 'package:flutter/material.dart';
import 'package:searchable_dropdown/searchable_dropdown.dart';

void main() => runApp(MyApp());

class MyApp extends StatelessWidget {
  // This widget is the root of your application.
  @override
  Widget build(BuildContext context) {
    return MaterialApp(
      title: 'Flutter Demo',
      theme: ThemeData(
        primarySwatch: Colors.blue,
      ),
      home: MyHomePage(title: 'Flutter Demo Home Page'),
    );
  }
}

class MyHomePage extends StatefulWidget {
  MyHomePage({Key key, this.title}) : super(key: key);

  final String title;

  @override
  _MyHomePageState createState() => _MyHomePageState();
}

class VagasDisponivei {
  String v_n;
  String v_id;

  VagasDisponivei({this.v_n, this.v_id});

  @override
  String toString() {
    return '${v_n} ${v_id}';
  }
}

class _MyHomePageState extends State<MyHomePage> {
  int _counter = 0;
  List<VagasDisponivei> _vagasDisponiveis;
  String vaga_name;
  VagasDisponivei selectedValue;
  void _incrementCounter() {
    setState(() {
      _counter++;
    });
  }

  @override
  void initState() {
    // TODO: implement initState
    super.initState();
    _vagasDisponiveis = [
      VagasDisponivei(v_id: "1", v_n: "abc"),
      VagasDisponivei(v_id: "2", v_n: "def"),
      VagasDisponivei(v_id: "3", v_n: "dgg"),
    ];
  }

  @override
  Widget build(BuildContext context) {
    return Scaffold(
      appBar: AppBar(
        title: Text(widget.title),
      ),
      body: Center(
        child: Column(
          mainAxisAlignment: MainAxisAlignment.center,
          children: <Widget>[
            SearchableDropdown(
              items: _vagasDisponiveis.map((item) {
                return new DropdownMenuItem<VagasDisponivei>(
                    child: Text(item.v_n), value: item);
              }).toList(),
              isExpanded: true,
              value: selectedValue,
              isCaseSensitiveSearch: true,
              searchHint: new Text(
                'Select ',
                style: new TextStyle(fontSize: 20),
              ),
              onChanged: (value) {
                setState(() {
                  selectedValue = value;
                  print(selectedValue);
                });
              },
            ),
            Text(
              'You have pushed the button this many times:',
            ),
            Text(
              '$_counter',
              style: Theme.of(context).textTheme.display1,
            ),
          ],
        ),
      ),
      floatingActionButton: FloatingActionButton(
        onPressed: _incrementCounter,
        tooltip: 'Increment',
        child: Icon(Icons.add),
      ),
    );
  }
}

code snippet of SearchableDropdown

if(widget.isCaseSensitiveSearch){
        print(item.value.toString());
        isContains = item.value.toString().contains(keyword);
      }