Swift binding to dictionary value

I have not tested your code but here's a few things I can see:

First problem

Okay first of all, I try to avoid when initializing the state properties like you are doing, with _accountBalances = State(initialValue: tempAccountBalances). Instead of this, I would pass the important data (accounts: [Account], date: Date) into the initializer, like you are doing, and initialize the accountBalances as an empty dictionary. Then in an .onAppear, or even .onChange(of:<accounts or dates>, initial), use the account and date to form the accountBalances. From my experience, the .onChange modifier is the best to use because it will refresh the accountBalances state variable, as soon as the view loads, as well as updates it when either of the account or dates change, which accountBalances is dependent on.

The pseudocode for this would look like this (I.e. it might contain a few syntax errors or incorrect labels, it's just from memory):

struct AddMonthView: View {
    ...
    @Binding var showAddMonth: Bool
    let accounts: [Account]
    let date: Date
    // Make this private so it cannot be passed in from another view
    // You are already initializing this as an empty dictionary
    @State private var accountBalances = [Account : Double]()
    
    // Don't need an init.
    
    var body: some View {
        NavigationStack {
            Form {
                ...
            }
            .navigationTitle ...
            .toolbar {
                ...
            }
            .onChange(of: (accounts, date), initial: true, action: updateBalances)
        }
    }
    
    ...

    private func updateBalances() {
        var tempAccountBalances = [Account: Double]()
        for account in accounts {
            tempAccountBalances[account] = 13 //defaulting to any number should work
        }
         
        accountBalances = tempAccountBalances
    }
}

Alternative approach

From the looks of it, you are needing this process of loading a accountBalances property from data passed in, just so you have something to bind the text fields to. Then, I'm guessing you need to synchronize the modified accountBalances with accounts in saveMonth to persist changes made by the text fields outside the view.

One other approach you can do is using a custom binding. I won't explain the exact syntax here, but I will refer you to a Swiftful Thinking video on this on YouTube -- it is very helpful and explains the syntax you'll see below, and more. But just know that with this method, you can bind the text field directly to Account.

struct AddMonthView: View {
    @Environment(\.modelContext) private var modelContext
    
    @Binding var showAddMonth: Bool
    @Binding var accounts: [Account]
    let date: Date
    
    var body: some View {
        NavigationStack {
            Form {
                List {
                    ForEach(Array(accounts.enumerated()), id: \.offset) { index, account in
                        HStack {
                            Text(accounts[index].name)
                                .frame(maxWidth: .infinity, alignment: .leading)
                            
                            TextField(value: Binding(get: { accounts[index].balance }, set: { new in accounts[index].balance = new }, format: .currency(code: Locale.current.currency?.identifier ?? "USD"))) {
                                Text("Balance")
                            }
                            .keyboardType(.decimalPad)
                            .submitLabel(.done)
                        }
                    }
                }
            }
            ....
        }
    }
    
    ...
}

#Preview {
    let date = Calendar.current.date(byAdding: .year, value: 1, to: Date())
        
    AddMonthView(showAddMonth: .constant(true), accounts: Account.sampleData, date: Date())
        .modelContainer(SampleData.shared.modelContainer)
}

I don't know of any reason your text field rows are duplicated, but with both these things I tested the code out and it seemed to run fine. Here is the code for your reference:

import SwiftUI
import SwiftData

struct Account {
    let name: String
    var balance: Double
    
    static var sampleData: [Self] {
        [
            .init(name: "Test 1", balance: 0),
            .init(name: "Test 2", balance: 0),
            .init(name: "Test 3", balance: 0),
            .init(name: "Test 4", balance: 0),
            .init(name: "Test 5", balance: 0),
            .init(name: "Test 6", balance: 0),
            .init(name: "Test 7", balance: 0),
            .init(name: "Test 8", balance: 0),
            .init(name: "Test 9", balance: 0),
        ]
    }
}

struct AddMonthView: View {
//    @Environment(\.modelContext) private var modelContext
    
    @Binding var showAddMonth: Bool
    @Binding var accounts: [Account]
    let date: Date
    
    var body: some View {
        NavigationStack {
            Form {
                List {
                    ForEach(Array(accounts.enumerated()), id: \.offset) {
                        index,
                        account in
                        HStack {
                            Text(accounts[index].name)
                                .frame(maxWidth: .infinity, alignment: .leading)
                            
                            TextField(
                                value: Binding(
                                    get: {
                                        accounts[index].balance
                                    },
                                    set: {
                                        new in accounts[index].balance = new
                                    }),
                                format: .currency(
                                    code: Locale.current.currency?.identifier ?? "USD"
                                )
                            ) {
                                Text("Balance")
                            }
                            .keyboardType(.decimalPad)
                            .submitLabel(.done)
                        }
                    }
                }
            }
        }
    }
    
}

#Preview {
    @Previewable @State var a = Account.sampleData
    let date = Calendar.current.date(byAdding: .year, value: 1, to: Date())
    
    AddMonthView(showAddMonth: .constant(true), accounts: $a, date: Date())
//        .modelContainer(SampleData.shared.modelContainer)
}