Can I use “You say Tomato, I (we) say X,” in the exactly same manner as “We agree to disagree,” in day-to-day conversation?

In the phrase you quoted, the implication is one of using two different words for the same exact concept, implying a difference where none exists. It is not simply about disagreement, and indeed implies the opposite: you are not in actual disagreement with each other, but both parties want it to appear that they are referring to different things anyway.

In the original song, the examples that were given are all of "upper class" pronunciations vs. "common man" pronunciations. For example, at the time, an upper-class American would be more likely to say "toh-MAH-toh", to distinguish themselves from the average person that would say "toh-MAY-toh". But both are words for the same thing -- a tomato.

In modern usage, the phrase has taken on a broader meaning of selectively using vocabulary to distinguish between otherwise identical concepts. In the Time example, depending on your particular political affiliation, you are more likely to call the government subsidizing of private companies a "bailout" (if you opposed it) or a "loan" (if you support it), but ultimately you are talking about the same exact action. The Time headline is meant to be understood as reading "Some say 'loan', others say 'bailout', but the result is the same"; they are using the reader's presumed familiarity with the more general phrase for effect.

In your example from Japanese politics, you are talking about a case where an actual distinction exists between the two options. There is a real choice between two possible actions, and the words you use to describe each of those options differ because they must. This is not a case where you would likely use the "You say tomato..." phrase.

If someone who opposed a tax increase began referring to it as a "penalizing the rich", for example, then it would be more appropriate to use that particular turn of phrase.

It needs to be two names for the same thing. In the example you gave, some say it's a "loan" others call it a "bailout"

But when it's a choice between two different things, "tax increase" or "spending cut" it's not a case of the same thing by a different name so it is not appropriate to use the "You say 'tomato'... " turn of phrase.

You say tomato , I say tomato (pronounced toMAYto/toMAHto) is a fairly common expression that basically means, "you and I both have our own opinions on this, and neither one is necessarily 'wrong' ". The Gershwin song may be rather old, but the expression is still in contemporary use.

As for using it in the context of Japan's current economic debate, the way you've worded it in your question reads a bit awkward. What works for a newspaper headline wouldn't necessarily work all that well in conversational English. However, if you were engaged in discussion with someone about which should come first – spending cuts, or the tax increase – and it looked like neither one of you was going to persuade the other to switch positions, a polite way to end the debate while the conversation was still on friendly terms might be:

You say tomāto, I say tomäto; I guess we'll just have to agree to disagree.