Is it possible to reduce the weight of still images (JPG or PNG) in the video (H.265)?

JPG and PNG can be reduced several times lighter by compressing (TinyPNG) at a level that is almost unknown to the public, but can you reduce the weight of videos like that?

Technically possible.

I would like to make H.265's video lighter, but in the case of video, is it correct to understand that the compression rate is almost determined by the codec method (not very close to software)?

No. Even with the same H.265 video codec encoder, the "compression rate" varies greatly depending on its software (=encoder) performance and parameter settings.

By the way, does the image quality vary depending on the software?

Yes. Rather, video codecs such as H.265 differ more depending on encoder software selection and parameter settings than static image codecs such as JPEG and PNG.