I heard it's never intended to be the same since initial rating for Lichess and chess.com respectively is 1500 and 1200. So they should have 300 rating difference on average. Quite fitting with what the other commenter claims actually.
ELO is relative. There's no reason why a GM ELO should be 2800 or 280 or 28000. So it's all decided by ELO of every other person. So if the ELO gain/loss calculation and audience of Lichess and chess.com are exactly the same, because of different starting position, I don't think they'd converge to the same ELO but instead will differ by starting position difference.
Also I can't really prove it mathematically but I guess average ELO would also hover on the starting ELO. Because I can't see why it would hover anywhere else and any ELO gained would be lost by someone else.
On further thought yes, I think you're correct.
When I started playing I believe chess.com let you select whether you’re beginner, intermediate or advanced and your start elo was based on that. Could be wrong, and it could’ve changed since.
I don’t think it would average out to a 300 elo difference simply based on the starting rating being 300 apart.
If everything else was the same, and people play enough games they will average out to the same elo.
The difference is caused by many factors. People don’t play enough games to sink to their real elo, the player pool is different, and you gain/lose fewer points per game with Lichess’s elo algorithm.