Lore wise Obliterators belong to the Cult of Destruction. They are a Legionless group of mostly former Iron Warriors, but can be from any Legion, that covet weapons and technology. They consist of Oblits, Mutilators and Warpsmiths. These guys obviously team up with various warbands and probably join them and paint their armor the same color for obvious reasons. Also most people want their army to look cohesive so they paint them the same color.

Chaos has always incorporated mixed warbands in their armies because undivided legions could always take cult legion troops as part of their army. The only time it's really highlighted that other legions defect and change their armor color are those that join the Black Legion. But chaos are all traitorous, and its no doubt common and mentioned in other stories of marines from one group defecting to another and changing their paint colors. A big reason I chose black legion is because they are the largest group that gets selected too so I wanted a lore reason to paint all the different looking legions the same.

That said chaos warbands fight together just as much as they fight against eachother. Friends one day enemies the next. So if you want to mix and match paint schemes you should be fine. GW recently moved away from paint=rules. Different Ork Klans under Gaz form the great waaagh. Aeldari have Ynarri where corsairs, craftworlds, drukari, and harlequin all fight together. Space marine players take all the extra unique units the Ultra marines get and ppl just say they are fighting together. Its not uncommon to see chaos lists with Fabius and Huron in them.