It's so much easier to move aroud, both for short term travel and longer term too. The common currency (at least in most countries) really helps as do things like no roaming charges. If you decide you want to go and look for a job in another country you just do it, no visa hassle or asking permissions.
I was born in the UK and moved to France (long before Brexit thank goodness) it would have been much harder without the EU.
I certainly consider myself a EU citizen, more than British or French (I now have both nationalities).
Of course it's not perfect but getting rid of individual coutry vetoes would help with a lot of things in the wider geopolicital sphere - and has already been done in many domains)