Well you're throwing an exception inside a try block, which will be caught, and your catch block then re-throws a different exception. You can either inspect the caught exception

async deleteUser(userId: string): Promise<boolean> {
    try {
        const userToDelete = await userModel.findByIdAndDelete(userId)
        if (!userToDelete) {
            throw new Error(`User with id: ${userId} does not exist`)
        }
        return true
    } catch(e) {
        if (e.message == `User with id: ${userId} does not exist`) throw e
        else throw new Error("Something wrong with the database")
    }
}

^{(Checking an e.code you put on the error with Object.assign, or testing for an Error subclass with instanceof, would be nicer than testing the message)}

or put the try closer around the await … statement whose errors you want to handle:

async deleteUser(userId: string): Promise<boolean> {
    let userToDelete
    try {
        userToDelete = await userModel.findByIdAndDelete(userId)
    } catch {
        throw new Error("Something wrong with the database");
    }
    if (!userToDelete) {
        throw new Error(`User with id: ${userId} does not exist`)
    }
    return true
}

which is nicer with .catch():

async deleteUser(userId: string): Promise<boolean> {
    const userToDelete = await userModel.findByIdAndDelete(userId).catch(e => {
        throw new Error("Something wrong with the database");
    })
    if (!userToDelete) {
        throw new Error(`User with id: ${userId} does not exist`)
    }
    return true
}