Most games from third-party developers still do not offer cross-play, this is because there are significant challenges involved for the developers as console architecture, business agreements, player networks, server locations, etc. are all factors that are considered. Officially right now, the Overwatch Development Team has stated they are interested in the idea of bringing Cross-Persistence, but that means only being able to share collected items on the same linked BattleNet account across all platforms, not actual multiplayer across their platforms.
Unfortunately, this is not the best place to provide suggestions for game features. The technical support forum is where players like myself and Blizzard Support Agents (who are different from the developers) to lend a hand for players who have technical issues (such as disconnections or game crashes) with Overwatch. Please consider posting your suggestion in the general discussion forum.
If in the event you are unable to post in the general discussion forums, please check this troubleshooting guide.