Note that immigration inside the EU is way easier than with other countries.
You don't need a visa, working permit, special bank account (for US citizens, since the bank must report income, etc, to the US gov), your driving license is valid, etc.
You change languages, currency, you need new bank accounts, new health insurance, completely different educational system. All administrations are different. The culture is also a shock. You will never find any of the grocery or shop you used to buy.
A French going to the UK is not much different than a Mexican going to the USA.
The only advantage is that you don't risk being deported by the police if they don't like you papers. You are effectively allowed to be there by the virtue of being European.
What's it with the bank accounts? You can transfer money within the EU without any problems. I pay my rent in Spain from a German bank account. Though, getting wired internet is another thing, these companies seem to ignore European law.
That's a huge advantage tho, especially if you don't already have a job offer at hand. In fact if you're not in a well paid/in-demand profession like software dev it is almost impossible to immigrate legally from outside the EU.
Not true for France. It might be easier for an EU citizen, but it is still a big waste of time. Getting a bank account still takes at least 3 visits. To hold a working card in your hand will take more then a month. Your driving licence is valid only for a year. And the Social Security nightmare is pretty much the same for everybody. And for the etc. prepare for that every bit will be a hassle.
And the parent is mistaking, he's probably American.
You cannot use a bank accounts well across EU country. Banks are a very localized service, they provide specific account number formats and payment methods by country. It will be a huge hassle day to day when you want a SEPA transfer and the guy expects a RIB or a sort code.
It's worse with UK and Switzerland because they have a different currency. You really need a local bank account or you get bankrupted by the fees and the exchange rate. Incidentally they are the only places where there are the big tech and finance companies to work for.
Most banks are national. There are very few banks that have an international presence, with decent service at the offices.
The last time I went to a French bank in a different region than where I opened the account, they told me they can't do anything and I have to go to my "home" agency.
I have UK and French accounts with the same bank. Both work fine for making payments in other EU countries. I can transfer money between them whenever I want (or spot a better exchange rate).
Using an Austrian bank account in Germany was basically indesitnguishsble from using a German bank account. Nevertheless opening a German bank account was trivial.
And all bank transfers within the EU (not sure if the U.K. is a special case) are free of charge, and IBAN/BIC used everywhere. I routinely transfer money between my Austrian and German accounts.
You don't need a visa, working permit, special bank account (for US citizens, since the bank must report income, etc, to the US gov), your driving license is valid, etc.