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
}