Informally, "k" is a common nickname for thousand but most currencies predate metric (and even SI). One interesting exception is "cent" - the revolutionary zeal in the new-born US presumed that the metric system would catch on, and the proposed pefix for hundreds - "centi-" - was adopted for the hundredth division of a dollar - a year or two, I think, before France adopted metric.
The dollar is broken into cents, not centidollars. The cent symbol is ¢ rather than c$.
The Euro had the opportunity to be a metric currency, but it isn't.
Dollar: Yes, that's right. This predated the formal adoption of the metric system, and the centuries of revision that have followed. The first version of the metric system, the system adopted by France, used 1s and hundreds (an "are" was (is still) 100 square metres, for example).
Euro: the Euro uses cents (or "cent", according to Brussels) as the hundredth division of the Euro). But since the US dollar every currency has moved to a 10^x system - GBP was the last hold-out so far as I know.