Hello,
I downloaded this file https://www.freepik.com/free-vector/bacteria-infection-virus-flu-background_7080504.htm from freepik. The EPS file opening properly on illustrator but opening as a image file only in Affinity Designer. I attached the EPS file.
It's probably due to the way that EPS file has been created and has been stored at all. There are several things which can influence the behavior of EPS files for other third party software, as there are for example ...
(see for example: Bitmapping 101: Troubleshooting EPS Files in Illustrator)
For EPS files which do use a lot of such Adobe Illustrator specifics, the Affinity apps can't parse and interpret the EPS contents in the same proprietary vector manner as Illustrator (the file creator) and so instead will only be able to showup the rasterized image (bitmap) portions of such EPS files.