OK, i'll take a swing at it, but in all honesty the real TL;DR answer to your question is "because it has little to do with politics, and is a reality of how language works and evolves, and a downside of English being a vague language".

Something opposite to a democracy, like in "US is a Republic, not a democracy"

That's actually not quite correct. More specifically, using incorrect terms.

What the opposition is, specifically, is of a Representative Republic ; vs "Direct democracy - in actuality, most sources would acknowledge that a Republic can be considered a form of democracy, not its opposite.

This Washington Post article gives a great overview with tons of citations to Founding Fathers, which I won't copy/paste here just to avoid clutter. But I strongly recommend reading it.

a synonym of conservative, as in "Republican"

That is simply a historical accident, in that, a large fraction of people who are following the ideology of conservatism (both in general, as an opposition to revolutionary change, and US-specific brand, which is strongly associated with Mills) happen to be in a Republican party for the last century; due to the fact that the Democratic party became a hotbed of "progressive" ideology (the original one, of Woodrow Wilson, and FDR), which clearly opposes conservativism.

For an extra dose of irony/confusion, Democrats and Republicans in USA were, once, the same exact party, "Democratic-Republican Party", which opposed Federalists.

Antonym of monarchy.

This concept goes as far back as Founding Fathers (at least. I'm not certain if the concept is far older than that), with James Wilson:

Then let us examine, Mr. President, the three species of simple governments, which, as I have already mentioned, are the monarchical, aristocratical, and democratical. In a monarchy, the supreme power is vested in a single person; in an aristocracy, it is possessed by a body, not formed upon the principle of representation, but enjoying their station by descent, by election among themselves, or in right of some personal or territorial qualification; and, lastly, in a democracy, it is inherent in the
people, and is either exercised by themselves or by their representatives.

OK, I'll take a swing at it, but in all honesty the real TL;DR answer to your question is "because it has little to do with politics, and is a reality of how language works and evolves, and a downside of English being a vague language".

Something opposite to a democracy, like in "US is a Republic, not a democracy"

That's actually not quite correct. More specifically, using incorrect terms.

What the opposition is, specifically, is of a Representative Republic ; vs "Direct democracy - in actuality, most sources would acknowledge that a Republic can be considered a form of democracy, not its opposite.

This Washington Post article gives a great overview with tons of citations to Founding Fathers, which I won't copy/paste here just to avoid clutter. But I strongly recommend reading it.

a synonym of conservative, as in "Republican"

That is simply a historical accident, in that, a large fraction of people who are following the ideology of conservatism (both in general, as an opposition to revolutionary change, and US-specific brand, which is strongly associated with Mills) happen to be in a Republican party for the last century; due to the fact that the Democratic party became a hotbed of "progressive" ideology (the original one, of Woodrow Wilson, and FDR), which clearly opposes conservativism.

For an extra dose of irony/confusion, Democrats and Republicans in USA were, once, the same exact party, "Democratic-Republican Party", which opposed Federalists.

Antonym of monarchy.

This concept goes as far back as Founding Fathers (at least. I'm not certain if the concept is far older than that), with James Wilson:

Then let us examine, Mr. President, the three species of simple governments, which, as I have already mentioned, are the monarchical, aristocratical, and democratical. In a monarchy, the supreme power is vested in a single person; in an aristocracy, it is possessed by a body, not formed upon the principle of representation, but enjoying their station by descent, by election among themselves, or in right of some personal or territorial qualification; and, lastly, in a democracy, it is inherent in the
people, and is either exercised by themselves or by their representatives.

OK, i'll take a swing at it, but in all honesty the real TL;DR answer to your question is "because it has little to do with politics, and is a reality of how language works and evolves, and a downside of English being a vague language".

Something opposite to a democracy, like in "US is a Republic, not a democracy"

That's actually not quite correct. More specifically, using incorrect terms.

What the opposition is, specifically, is of a Representative Republic ; vs "Direct democracy - in actuality, most sources would acknowledge that a Republic can be considered a form of democracy, not its opposite.

This Washington Post article gives a great overview with tons of citations to Founding Fathers, which I won't copy/paste here just to avoid clutter. But I strongly recommend reading it.

OK, i'll take a swing at it, but in all honesty the real TL;DR answer to your question is "because it has little to do with politics, and is a reality of how language works and evolves, and a downside of English being a vague language".

Something opposite to a democracy, like in "US is a Republic, not a democracy"

That's actually not quite correct. More specifically, using incorrect terms.

What the opposition is, specifically, is of a Representative Republic ; vs "Direct democracy - in actuality, most sources would acknowledge that a Republic can be considered a form of democracy, not its opposite.

This Washington Post article gives a great overview with tons of citations to Founding Fathers, which I won't copy/paste here just to avoid clutter. But I strongly recommend reading it.

a synonym of conservative, as in "Republican"

That is simply a historical accident, in that, a large fraction of people who are following the ideology of conservatism (both in general, as an opposition to revolutionary change, and US-specific brand, which is strongly associated with Mills) happen to be in a Republican party for the last century; due to the fact that the Democratic party became a hotbed of "progressive" ideology (the original one, of Woodrow Wilson, and FDR), which clearly opposes conservativism.

For an extra dose of irony/confusion, Democrats and Republicans in USA were, once, the same exact party, "Democratic-Republican Party", which opposed Federalists.

Antonym of monarchy.

This concept goes as far back as Founding Fathers (at least. I'm not certain if the concept is far older than that), with James Wilson:

Then let us examine, Mr. President, the three species of simple governments, which, as I have already mentioned, are the monarchical, aristocratical, and democratical. In a monarchy, the supreme power is vested in a single person; in an aristocracy, it is possessed by a body, not formed upon the principle of representation, but enjoying their station by descent, by election among themselves, or in right of some personal or territorial qualification; and, lastly, in a democracy, it is inherent in the
people, and is either exercised by themselves or by their representatives.