Well technically there are different trading hours, calendars and sessions for each market / segment of NYSE. In practice though I don't think you will find many companies listed on multiple subsegments of NYSE in 2025.

Roughly speaking, you will find the prestigious / big market cap / rich companies listed on NYSE main market, ETFs on NYSE Arca, small caps on NYSE American (ex Amex), and very small caps on NYSE National.

The listing requirements and prices, as well as fee structure also differ for each market. National has a fee structure to incentive adding liquidity for instance.

If the asynchronicity bothers you, imagine that you can also trade e.g. HSBC secondary listing on NYSE NY market hours + the ADR of HSBC HK on NYSE NY hours + HSBC primary listing on LSE on London hours + HSBC HKSE on HK hours for a lot of fun.