Actually, the answer is yes to both questions.
Regarding rivalry, there’s the classic “literally any canine pokémon versus literally any feline,” as well as “literally any feline pokémon versus literally any rodent.” Granted, few of these rivalries actually have any current truth to them. When it comes to canines and felines, that simply has roots in both of their social behaviors. Dog pokémon are highly social and expressive, and they display affection through non-combative play routines. They also have a tendency to chase after pokémon that are smaller than they are, either to hunt or to play-hunt, depending on whether or not the dogs in question are wild. On the other hand, feline pokémon are more reserved and play by sparring with one another, and they don’t take well to anything deemed as a threat, even if that threat is larger than them. Thus, a dog pokémon may attempt to chase after a cat pokémon, which in turn is interpreted as an act of aggression to the cat; alternatively, the cat pokémon may attempt to play in its own way with the dog, which results in the act being interpreted as aggression by the dog. Either way, these miscommunications are interpreted by humans as bitter rivalry, rather than as what they are (that is, an unfortunate crossing of perfectly natural behaviors).
As for felines and rodents, well. Considering the fact that feline pokémon eat many creatures smaller than them, including and especially rodent pokémon, it only stands to reason that the two don’t always have positive interactions for both sides. Granted, humans didn’t help in that regard either: as rodent pokémon had been pests to us at one point, we’ve trained feline pokémon, their natural predators, to … well, chase them off.
Of course, these are only perceived rivalries, or merely interactions between species that are not combative but rather pokémon simply following instinct, no animosity intended. On the other hand, there are plenty of instances of actual rivalries in the pokémon world. Both types of basculin, for example, if only because red and blue basculin are highly territorial and protective of their own kind (and thus, yes, a little bit racist). Then there are yungoos and Alolan rattata (intended to be predator and prey and instead ended up as rivals for territory and resources), braviary and mandibuzz (territory disputes, usually), heatmor and durant (for the same reason seviper and zangoose don’t get along), gyarados and anything that moves, and for some reason, clefairy and jigglypuff. Some even say omanyte and kabuto are rivals, but this is more hearsay from obscure texts, rather than anything founded on scientific fact.
As for symbiotic relationships, that can actually apply to a number of pokémon. Many pokémon are capable of living in harmony with one another, within delicately balanced ecosystems. Take the bibarel, for example. Bibarel create dams, which in turn create freshwater lakes for finneon and shellos. Thus, finneon and shellos tend to gather and follow (in rivers and streams, of course) wherever bidoof and bibarel are, just waiting for either pokémon to construct new habitats and breeding grounds.
Then you have the more well-known symbiotic relationships, such as shellder and slowpoke (the two evolve together to form slowbro or slowking), mantine and remoraid (remoraid helps mantyke to evolve, in exchange for protection), and shelmet and karrablast (they help one another evolve, even in the wild). Most lines that are considered counterparts to each other (nidoran, plusle and minun, miltank and tauros, volbeat and illumise) are likewise actually symbiotic species, as they gather in pooled herds to use their lines’ respective skills for survival. For example, although they may mate with other pokémon in a forest in order to obtain fertilized eggs, nidoran herds will set up combined nests guarded by nidorina and nidoqueen (which are better at defending territory). The male counterparts, nidorino and nidoking, will often either establish and patrol outer boundaries of the warren or go out to hunt and forage for the group, as, being the more aggressive of the two, they can form dangerous mobs to effectively drive away predators before they even reach the nests.
