A similar atmosphere where the story focuses on the relationship between the characters, members of the same club in their school.
The main character (Kumiko / Hachiman) questioned the relationship and try to resolve disputes between the characters. (In OreGairu there is more romance)
Additionally we follow the story from the viewpoint of the main character (Kumiko / Hachiman) and have access to her/his thoughts and impressions.
The secondary characters are also important in the two series: for example, Asuka (Euphonium) and Hayato (OreGairu) are friendly appearance but seem to hide their true personality.
The main difference is that Euphonium is an anime about music unlike OreGairu.
If you like the series that showcase the characters, I recommend both.