My first son has dual citizenship one EU, one other country not elegible for ESTA but currently holding a paper visa valid for 5 years from a previous travel when his EU passport was not available yet. We will travel again, this time with new baby brother who has EU passport and ESTA ready. Is it possible to request a second visa (ESTA this time) for my first son on his EU passport eventhough he already holds a visa on another passport? The main reason to do this would be to have as many EU passports in my family as possible when entering the US to avoid unnessesary scrutiny and trouble with border control.
You are a dual nationality traveller, and you want to know if your EU-member-issued passport is eligible for the US Visa Waiver Program and thus applicable for an ESTA
So long as you are not a national of, or recently visited, a country on the exclusion list (Iran, Iraq, Libya, Somalia, Sudan, Syria or Yemen - there may be others), then you are eligible to apply for an ESTA on the EU-member passport, if the EU-member nation that issued the passport is eligible for the Visa Waiver Program, as not all EU-member nations are.
That is important enough to repeat: not all EU-member nations are eligible for US Visa Waiver, and thus not eligible for the ESTA program.
I have dual citizenship from both a VWP country and a non-VWP country?
If you are flying under the terms of the VWP, with your VWP passport, you must apply for ESTA. If you are using your non-VWP country's passport, you will require a visa and therefore ESTA does not apply to you.
During the ESTA application process, you will be asked about dual nationality - you should answer these questions truthfully and to the best of your knowledge.
Its also worth noting that the presence of EU-member nation passports in your group will not stop individuals being considered under their own credibility at the US border - it should not sway a border official if your child has an EU-member nation passport and you do not, when your entry is being considered.