At the time, competition with Steam wasn't really the issue. Steam was, and is still, considered more or less niche for chatting, except among people who actually use it regularly. Even in gaming, most people who had a dedicated need for a messaging/voice application used Skype or Teamspeak or Mumble or something. So when Discord soundly replaced those, it seemed like it had more or less cemented its position.
However, these days, Steam seems to be trying to mimic Discord's look and feel, improve its voice calls, add better group chats, and generally try to take back some of that territory. They're not the only ones who seem to resent the way Discord's taken over the chatting needs of their user-base (I noticed that even Reddit, one of the biggest hubs I know of discord servers, seemed to be trying to make its own chatroom system), but what's interesting is that Discord seems to be replying in kind. Recent developments on Discord seem to have focused on improving their Overlay to a workable state, (no easy feat). and adding a game-launcher. I've also heard that they might make a selection of games available to Discord Nitro subscribers, possibly paving the way for Discord to even act as a games vendor. The days in which it seemed like Steam would roll over and let its chatting functionality wither and die seem to be long gone, but it may be that Discord will grow stronger yet in response.
Though it seems ridiculous that these two largely-unrelated services would be engaged in any kind of competition, it seems to be the nature of modern capitalism that every business must expand indefinitely and attempt to monopolize more and more functionality, lest it be consumed. Maybe one of these services will even completely replace the other. Now wouldn't that be a thought?