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Why should Sauron notice only Frodo?
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The answer lies in one of your quotes-
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And far away, as Frodo put on the Ring and claimed it for his own, even in Sammath Naur the very heart of his realm, the Power in barad-dur was shaken...
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Take note of two things- 1) he claimed it in "the very heart of his (Sauron's) realm" and 2) he did it as he "put on the Ring".
That's why Sauron noticed immediately.
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According to the text, you would then have to believe that he was literally his entire realm. Because, a tremor ran throughout his realm in addition to the trembling of the tower.
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This is somewhat true. Though Sauron isn't his realm, his will and power can be found driving and binding together everything in his realm.