Does iOS's native image editor lose image quality?
Yes, that is the case. When you edit a JPEG photo and save the result as a JPEG image, it will be re-encoded and thus you'll suffer some loss of quality. The only way to avoid that is to save the photo in a lossless format (for example RAW). You cannot do that with the built-in Photos app.

You can find other apps that allow you to save both the original photo and edits in better quality - like for example PureShot and others. Even though the resolution is the same, different apps have different quality settings in regards to how much JPEG compression is applied.

Note that even though the original photo has low compression (and high quality), just editing it in the native Photos apps will save a new JPEG with higher compression and lower quality. For example read this article where they loaded a 5.59 MB photo, applied an edit and saved - and got a 1.49 MB photo out of the native app:

Editing App Comparison

Another factor when comparing editing apps is to ensure that they perform all your edits against the original photo before exporting a new JPEG. If they create a new JPEG for each edit and use that as the source for the next edit, you'll experience the quality degradation multiple times.