There was certainly plenty of competition on the right. But I remember McManaman as the type of winger who was adept on both flanks and often at Liverpool he had a a floating role across the midfield (they often played a 3-5-2). He was a vital part of the Madrid side that won the Champions League in 2000 as part of a midfield three with Redondo and one other. United had an incredible midfield at that time yet they were really blown away by Madrid that season. And he was an influential player throughout his time there. In the three seasons ahead of the 2002 World Cup, he played 47, 42 and 38 games for Madrid - and that was competing with Figo and Zidane for a spot in midfield. Eriksson not taking him in 2002 was nothing to do with versatility IMO and simply about the Swede not rating him - which always seemed a bit odd given how well he was doing in a variety of midfield roles for a club that had just won its third Champions League title in five seasons.
I was still in primary school and early secondary school at that time with no Internet access, so my memory is a bit sketchy.
What I remember was that Real Madrid put him all over the place across the midfield during the start of the Galaticos era (which was close to the 2002 World Cup) and he did not have a designated role in the club. I think what I meant that he was not a versatile player is that players at the time had a fixed natural position, and could play in other positions but were not effective at all, unlike today's players like Mikhi, Hazard or Sanchez who can play at left, center and right forward positions. I guess you could say that he was versatile for his era and he is a pioneer of what it means to be a versatile utility player.
I just checked on transfermkt and I found out that in the 2001/2002 season after Zidane came, he only started 14 La Liga games. He started quite a number of games in 1999/2000 and 2000/2001 but found himself gradually phased out under the Galaticos policy. I think like one of the posters said, Eriksson was quite stubborn with his 4-4-2 with Heskey and Owen upfront. For my CM and FIFA games, I usually selected McManaman for the England squad to do what he did at Real Madrid, that was to come on as a sub and fill roles across the entire midfield. In my opinion, I think the versatility meant that he did not start many games and was constantly thrown on as a sub or utility player, and Eriksson wanted players who were more specialized or played more often, which was why he took Scholes, Lampard, Gerrard, Becks, Butt and Sinclair. Because of this insistence, he actually went to the World Cup with no left winger.