Sorry, I missed the part where you mentioned living in the EU. One of the siblings comments has a good explanation for ITINs. My wife used hers to open a Vanguard account despite having no SSN at the time. She did have a visa and a US address though, so I'm not sure how it would work for you (and if it would be worth the trouble; probably not).
I'm a French citizen, and wish I could help you regarding investing in the EU, but unfortunately I only got interested in it after moving to the US. Back in France it seemed like most people invested in real estate rather than in the markets.
Thank you anyway, I have a friend who's knowledgeable and he helped me find the tickers for the EU versions of the Vanguard ETFs. As I understand it, I won't need any tax presence in the US to buy them (e.g. https://www.bloomberg.com/quote/VWRL:LN), so I should be set. I'm trying to open brokerage accounts now, and it seems like everything should be simple after that.
I'm a French citizen, and wish I could help you regarding investing in the EU, but unfortunately I only got interested in it after moving to the US. Back in France it seemed like most people invested in real estate rather than in the markets.