History doesn't have a direct reflection on the club's performance it will only give the experience on the pitch, but both real Madrid and Liverpool have multiple records of champions league winning so I will not concentrate much on history but rather on the clubs' performances.
It's true it's just a record that becomes more motivation for the players to see their record in the past. But for performances it depends on their current form (when they play the game). But sometimes history affects the mentality of players which ultimately affects the final result. For example, team a is very difficult to beat team b, at the same time that difficulty is also difficult for them to break.
indirectly indeed records do not really affect the players in playing, but there are times when records give a team pride to continue to be able to maintain that record and coaches will usually make past records as motivation for players to continue to be able to achieve victory, and true as stated said by @maju69, sometimes the guts of a team will shrink because they really think if the other team is better than them and that will make their mentality down.