From the way the names are formulated I would guess that is not going to be the case.
When they are named after elemental ideas (water, fire, earth, electricity, wind and so on) you often instate the rock paper scissor system.
Whereas when they are named after the effect (burn, shock chill/cold) it is often paired with a resistance system. For example resist cold or resist shock.
And I conclude, that it will be important to “raid” titans or other epic bosses in groups, which are well combined in “elements”. Since here is no focus on pvp
The problem is that burn counters chill and chill counters burn they are opposing forces in opposition to pokemons fire burns plants that absorb water that can dose fire, in that case the forces are stronger/better against the other force but burn vs cold is going to be all about which one have the most power.
Cold coutners fire though as a cold flame is a dead flame, and the water ice have as an element counters fire too.
Fire is reliant on heat so therefore ice/cold kills it.
Heat is directly countered by an equivilant or larger amount of ice/cold
Cold is directly countered by an equivilant or larger amount of fire/heat
Ice is reliant on cold so therfore fire/heat kills it.
Why would damage types compete with each other?
I think that elemental creatures will just be immune to the damage type they are inflicting (just like in hundrets of RPGs before )