According to Wikipedia, the UEFA Champions League has been held annually since the 1955–6 season. But the Utilita Football Yearbook (formerly Rothmans) lists 1955 to 1992 as the European Cup and 1993 onwards as the UEFA Champions League.
Real Madrid have won the competition(s) 13 times – six times as the European Cup, and seven times as the Champions League.
No other club comes close. Milan have won it seven times – four times as the European Cup and three times as the Champions League. Bayern Munich and Liverpool each have a total of six wins, and Barcelona have five.
Because the number 13 is given as a footnote to the answer, we know that what this question means is "Which club has won the European Cup and the Champions League most times, in total?" But the question itself is ambiguous; it could be taken to mean which club had won it most times when it was the Champions League.
Whichever way you take it however, the answer would still be Real Madrid. Even if it had asked which club had won it most times when it was the European Cup, the answer would still be Real Madrid.
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