Assume I have a Roth IRA account and a mutual fund account through Vanguard, and that both of these accounts invest in the same fund. Does the minimum fund balance apply to the Roth IRA and mutual fund separately, or would their summed balance be the sole consideration?
The Vanguard minimum applies for each fund, within each account. So, if as in your example you have a Roth IRA account as well as a "regular" taxable account, each account is treated separately, even if you happen to choose the same funds to invest in within each account.
This isn't explained quite as clear as one might hope, but their website and prospectuses talk about the minimum being for a "fund account", and each fund account is separate.