There are different rules for every league, but broadly, while you can own clubs in multiple different countries, it is not possible to have a controlling interest in two clubs playing in the same competition.
In the Premier League, shareholding in a second club is restricted to 10%, which means it would not be possible for the Saudi sovereign wealth fund (PIF) (or, one presumes, its chairman, Crown Prince Mohammed bin Salman) to buy more than 10% of Manchester United, since they already own Newcastle United.
UEFA's rules are slightly less strict, though. Individual people or legal entities are permitted to own any "non-decisive influence" shareholding of a second club, without breaching the rules. "Non-decisive influence" here means owning less than 50% of the shares. So you could theoretically own 100% of Manchester United and, say, 25% of Juventus, without causing any problems, even if both clubs were in the Champions League.
This is why RB Leipzig and RB Salzburg were both able to compete in the Champions League - no single person or legal entity owns more than 50% of both of those clubs - you may recall that there was initially something of a furore about these two clubs competing in the same competition in 2017-18, and the matter was eventually referred to the Club Financial Control Body adjudicatory chamber, who eventually concluded that a series of “significant and substantial” changes made by the clubs (basically, a load of senior Red Bull execs at Salzburg, stepped down) ensured that they fulfilled UEFA’s eligibility criteria.