If you are a US person and you control certain foreign financial accounts worth more than $10,000, you must report them using the "FBAR" report. This is a reporting requirement only, no tax is owed. I don't know if a mutual fund must be reported, you can look here for the rules and a link to the report.
https://www.irs.gov/businesses/small-businesses-self-employed/report-of-foreign-bank-and-financial-a...
As far as taxes are concerned, you pay US tax according to US law. US law does not tax unrealized gains.
Now, that can get a little tricky depending on what you own and how it works. For example, dividends may be taxed as ordinary income or as capital gains. And if the dividends are reinvested to buy more shares, that can be a taxable event that increases your basis in the fund, or it could be a non-taxable event that does not raise your basis (but results in more tax when you withdraw because you have more shares with a lower average basis). So I don't feel 100% confident in saying you owe nothing until you withdraw funds.
I don't think your advisor is correct about calling these custodial accounts.
I also don't see where form 8621 comes into play.
@pk
