First of all, unless you're violating copyright or patent law, I wouldn't waste 2 seconds worrying about being seen as having "stolen" an "idea".

With that said, consider this: your product is not the sole basis for competition in the market. Your product is also not (hopefully) static and unchanging. And also, "the market" for your product can most likely be segmented in many ways, some of which would likely allow you to avoid head to head competition with this other company.

My recommendation is that you read the book The Discipline of Market Leaders which does a good job of explaining the ideas behind different vectors of competition: product, price, operational efficiency, customer intimacy, etc.

Is it even possible to focus on Europe whilst the US company (hopefully) focuses on the US (for now)?

Absolutely. In fact, if the other company focuses on the US to exclusion, you could focus on Europe, Asia, Africa, South America, etc. and "encircle" them. One of the big Japanese heavy equipment manufacturers famously did something like that to Caterpillar. For more on that, see:

https://www.google.com/search?num=30&q=komatsu+encircle+cate...