There is only one fundamental requirement: that the company's actions be anti-competitive. It is not necessary for the company to have a monopoly or even a majority share of a market (though these are factors). Furthermore, ownership of the platform is a significant antitrust factor because it drastically ups the risk and concern for anticompetitive behavior. (The Disneyworld analogy brought up below does not apply. Not only does it mix up antitrust law with property law, it ignores the crucial distinction between Apple and Disney: Apple openly invites others to participate in commercial activity on their platform, whereas Disney does not.)
Standard Oil, AT&T, and Microsoft may be the marquee cases, but they're not representative cases of the extent of antitrust law.