A quick entry, just about common rivalries between species.
Underground animals like moles do NOT like Avians like birds. (not only animal ancestry but lifestyle differences too.)
Mammals do NOT like sea-creatures and vice versa. The fact, that they have to take care in construction and also preserve and protect oceans via law bothers a good chunk. + Sea animals that also have animal ancestry (like bears with fish and fish with bears.)
Seedrians do NOT like Mobians. Mobians actually do not mind Seedrians. Seedrians frown down upon mobians, due to their evolution being similar to that of a human. Mobians are aware of that fact, so are aware of the discrimination against them, yet still do not obviously take it. Certain hate-groups have raided the planets before and tried to wage war, but those groups are frowned upon by both parties usually. Some outliers still believe in the fact, that seedrians are humans.
Said outliers, are the ones that look down upon seedrians instead.
Although the two planets have obvious rivalries between one-another's species, they still allow them to visit and travel, so long they consent to a heavy-inspection of luggage and belongings, as well as psychological test they need to fill out, to determine their stay.
Insects do not like Avians, again. Avians and sea-creatures are frowned upon the most.
And even some animosity exists between the species themselves:
Groups such as foxes are disliked by bunnies, alligators dislike hippos, etc, etc. Anything from what we know from the real-world, that has natural predators, will be real grudges they hold against each other.