I searched a lot, but only found the ICE-1 on demo tour in North America in 1993.
There however was the Danish IC3 (silly thing about easily confused names, did Piko fall for this, or was it a "toy" train idea?), a totally different train from the ICE3, that was tested by VIA in Canada, and returned, see rolling stock for VIA in wikipedia.
The IC3 wikipedia entry (NOT ICE3, just IC3) itself also lists Amtrak as a former operator, so they probably tested it as well.
So it looks like that was a totally different train though, but those Piko cars appear to indded loosely match the DSB IC3, which was actually a self propelled multiple unit, so relationship to the ICE3.
As far as I know, the only German Siemens ICE type ever run in North America was the original ICE-1, that toured USA in 1993, but was not bought.
I stress Siemens here, as Bombardier in the late 90s and early 2000s had bought out several old German locomotive smiths, and was producing high speed locos, multiple units, cars, etc in Germany, so the Acela still has a lot of German engineering, but is not officially in the ICE family.
Having said that, the ICE family has roots in the 80s, ICE3 dating to 1999, and they were built by consortiums. The individual consortium members were varied, and there were once about 1/2 dozen or so railway equipment makers in Germany alone. So, the companies that Bombardier absorbed were all likely involved in ICE building!
It is all a bit messy, but the Piko train in Amtrak dress ... right now seems to be a fun toy, and that is what our little trains are after all. At age 59 and after 55 years with model trains, I am not past enjoying a fun fantasy machine as an escape in a fantasy land :)