Multiple pixels or single pixels used on multiple stores are a very common method used to prime pixels and begin feeding "Purchase" website conversion data to that pixel.
This is a very common method used in E-commerce, especially those that use custom themes on sites like Shopify to build up stores and sell them on to buyers with its own pixel id.
They then can transfer the other pixels that have been fed data to replicate the process with new stores as they have accumulated buyer pixel fires, and the new Facebook ad's or targeting will be more trained when going out looking for buyers or retargeting old buyers on the newly created website.
Overtime if you kept doing this you would obtain a massive data pool of buyers if you used multiple pixel id's on all the sites generating sales.
Using a single pixel for multiple sites will see the data diluted across all the websites hosting that pixel to track conversions so It would be important that if starting out this way the websites hosting the pixel had similar lines of targetting or already had buyers going through the funnel on a daily basis to train that pixel quickly for "Purchase" or specific "Conversion" fires seen across all these sites. Also note that because the pixel would be hosted on different websites, the funnel may also differ per website, this can effect greatly the data collection and could cause issue when retargeting and optimising ad's or campaigns for certain Web Conversion types.
In the end its all about data with Facebook, the more you have especially for buyers or add to carts the better.
So simple answer is yes, you can have One pixel across multiple websites/shops. But the real question is do they all follow a similar path when it comes to converting visitors and is the conversion goal the same on each?
Regardless there are still ways to have a single Pixel Per Website but also include a separate pixel installed on all domains that collects conversion data as a whole.
