Is there a standard video logo size? When could a 1500px png logo 300 ppi size be possibly needed?
First lets talk about the PPI/DPI thing. Its entirely irrelevant for the context of video production. It literally never comes to play.
Second a full hd aka 1080p image is 1920 x 1080 pixels in size so ideally youd have a version of the logo thats designed exactly for those pixel dimensions, with whitespace and all. But 4k consumer (4k UHD) devices are 3840 × 2160 pixels, and cinema (DCI 4k) devices are 4096 × 2160. Not to talk about 8k stuff.
So for any modern video producer a 1500 pixel logo seems a bit low resolution indeed. But why they wouldnt use the vector files beats me?
So to directly answer the question:
When could a 1500px png logo 300 ppi size be possibly needed?
Every time they need a full screen logo. So pretty much at the start of each video.
You are not thinking in the correct context.
A video has a specific pixel resolution if you want to specifically target an optimum display.
For example FullHD has specific dimensions of 1920x1080px. So, define your logo on how it should be using a canvas of that specific size. It does not matter the pixel size. That pixel size is how you want it to be.
Is there a standard video logo size
Thank god no. Why would anyone want a standardized look on a logo on a video? Make the logo any size you need or want.
But something to take into account is that some video edition programs can use either a vector logo file or a raster one.
If you want to have a logo on a higher resolution screen, make it bigger than the one needed for HD, so you do not see a pixelation or use a vector version and define whatever size during editing.