As recommended, I customised Photonic to have a specific Shortcode in order to avoid a conflict with Meow but this was not enough.
The shortcode and the lightbox are two different things. The shortcode runs on the server and will fetch your photos. The lightbox runs on the browser, and will show your photos in a specific way.
Meow Gallery also uses the gallery
shortcode, and sometimes has the same parameters as Photonic. That causes a conflict and prevents Photonic from running on the server, which is why I offer the custom shortcode functionality. But once the photos are in the browser, the shortcode has no impact.
Within Photonic the lightbox specifically looks for Photonic’s galleries. That is why Photonic’s lightbox does not affect other photos. Other plugins too can use the same concept and code their lightboxes specifically to work for their galleries instead of all galleries.