I've lived in London for 1 year, and Paris for 4 years (density: 55,000 people/mi², almost like Manhattan). You don't actually interact with others. You just cross their paths. And most of them are strangers you'll never encounter again. Or if you do, you won't notice it.

I guess I've become accustomed to the fact of being constantly surrounded, at almost any time of the day or night, although this feeling of permanent busyness is not exactly tangible.

As long as you remain aware of the fallacy of these interactions and don't expect anything from them, you're fine.