Master of Ring by Ring mastered?

[the phantom]

Why should Sauron notice only Frodo?

The answer lies in one of your quotes-

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...

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.

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.

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.