I've ran into this issue when trying to load a new logo for my Atari Gamer website into a watermarking app I use on my iPhone. In the past, all I had to do was drop my PNG into Photos on the Mac and it would come up, transparency and all, on my iPhone.
With iOS 13, this happens...
The image is first loaded correctly with a transparent background, then a few seconds later it changes to have a white background instead.
When opening the image in Photos as soon as it shows up, it does indeed show up with a transparent background. This isn't always consistent though, opening and closing the image several times in a row shows it either with a transparent background or with a white background, but after a time it always shows up with a white background.
Unfortunately using an image like that in a watermarking app becomes impossible as it always has a solid white background. It appears (from what I read on various forums and support websites) that this was reported to Apple as a bug multiple times, however Apple still hasn't fixed this problem.
Have you found a solution or workaround?