Therefore, it can't possibly affect your chances to win like it does in dice. It is 50/50 in terms of winning or losing for two players before the cards have been dealt (if we discard the ramifications of skill and expertise from the equation)
A house edge does not only mean a certain amount of advantage of the casino over the player in terms of winning chance. In a dice game, your winning chance could also be set at 50%, which means the dice site does not have any amount of edge over you in winning probability. That means that "It is 50/50 in terms of winning or losing" just the same. But the house edge is in the payout or winning prize. Your chance of winning against the house may be 50:50 but your winning prize will not be x2. That means the house still has the edge over you
You know what they say?
They say that we should compare things in equal conditions (otherwise known as "all other things being the same"). More specifically, to make dice look like poker (for two players), you should set the payout to 2, which would match the outcome of a poker game (what you win is what the other party loses). And in that case, your chances won't be 50/50, but rather something like 49.5/50.5 in favor of the casino. Basically, we have just excluded the payout factor from the equation and made things equal between dice and poker
In a poker game, you are playing against another player but a portion of the winning will go to the house. In which case, the house wins every time a game ends, regardless of who wins. This is also the same rule that applies over sports betting. That may not be aptly called house edge but it is somehow similar to that
I wouldn't call that "somehow similar to that". The reason is quite simple. You are not playing against the house, so even if it does somehow look similar to the house edge, it is not a house edge in and of itself (which follows straight from your post, by the way)