Your elo does go up or down. It's your rank that doesn't go up or down.
Elo is a hidden number that you cannot see in any way, which changes based on matches.
Your rank is based on your own elo as well as the elo of all other players in such a way that the distribution of ranks is approximately normal. So, for example, if there's 100 people in Initiate I with 50 elo, assuming 50 is the lowest elo anyone has, and 100 people in Initiate II with 100 elo, and no other changes, then if everyone in Initiate I raises their elo to 100, then they stay in Initiate I, and the 100 people with 100 elo are dropped down to Initiate I, because apparently 100 is now the lowest elo anyone has (and the rank system ensures that everyone with the lowest elos is in Initiate I, regardless of which number that is). Your rank is based on the global distribution of elos, so what other players are doing, other players you don't even see in matches, affects your rank. Another example is that people with the average elo across all players go to Emissary III.
Did you and your friend have different ranks or the same rank before this happened? If it's the same, that would be kind of weird. But if it's different, that's completely possible and normal. Especially if you both win and lose.
Also note that there's around 5000 people in each rank, so worst-case there's about 2500 people you need to "pass" before seeing any difference.