[SalesForce] Lightning currency input component that respects the records IsoCode

Short answer: you can't do this without a major investment.

Longer answer: my job is developing FinancialForce's PSA application, which requires supporting dated exchange rates and other ACM features on custom objects. Salesforce has confirmed to us more than once that there is no native way to do this, so we've implemented our own currency conversion from the ground up to support our needs. We do support reading the dated exchange rates from ACM's tables, but we still have to do all our own math on the conversions in apex logic. At the very least this means that you can tap into the same currency data for your calculations, even if the implementation of the logic must be done independently.

The net effect is that if you want ACM functionality outside of the officially supported objects you're going to have to build you own currency conversion system, and your own apex-managed rollups rather than using the native functionality, for this to work. If you do go this route you'll probably want to turn off parenthesized currency conversions as they'd use the non-dated rates and effectively lie to users of your custom currency system.

This is obviously a non-trivial solution to the problem, but it's the only one that's currently possible today.

It's not really an answer to original Q but it's too big to fit into comment. And you've made a follow-up comment asking about it so here goes...

(note: I'm lucky and I didn't had - to date - to work in an org with dated exchange rates. Added layer of complexity lurks there)

If you're using a rollup summary of currency fields - magic is done for you and all is fine.

If you're calculating the summary manually in a trigger or batch job - roughly speaking you have 3 ways to choose from:

Naive:

SELECT AccountId, CurrencyIsoCode, Amount
FROM Opportunity

This is the worst way. You either ignore the currency and add apples to oranges or you have to manually examine CurrencyIsoCode and convert in each row. Your rounding errors will accumulate, good luck with that.

Convert to user's currency

SELECT Id, AccountId, convertCurrency(Amount)
FROM Opportunity

Then you loop and make a fairly straightforward summary. Maybe you don't add apples to oranges, maybe you're adding bananas - but they're all bananas so who cares. You'll perform only 1 currency conversion on the total result and all will be fine in the world.

You might consume lots of script statements to roll them up but convertCurrency is great to "just get stuff done" when all you want is to display something in Visualforce in the way the user prefers.

Convert to organisation's currency

SELECT AccountId, SUM(Amount)
FROM Opportunity
GROUP BY AccountId

Similar to previous. Bit more fuss to access AggregateResult but less code statements. Same amount of SOQL rows is wasted (the 50K limit) but GROUP BY has extra limit of up to 2000 rows in resulting set - so if you're recalculating it in batch you might have to tune down the batch size. Didn't test it but at least in theory should perform better as you throw the heavy lifting task at DB (less memory used, less network traffic etc).

As for the conversion rates (applicable to all methods although in scenario 2 and 3 you'll multiply only once) - I use something like that:

Map<String, Decimal> conversionRates = new Map<String, Decimal>();
for(CurrencyType ct : [SELECT IsoCode, ConversionRate FROM CurrencyType]){
  conversionRates.put(ct.IsoCode, ct.ConversionRate);
}

Then I just access the map using parent's (so for me - Account's) CurrencyIsoCode. Most of my code is using the #3 method so it's a straightforward multiplication. In worst case scenario for #2 you'd have to go user's currency -> org's currency and then org's currency -> account's currency.