Does the motto “Don’t tread on me” have a specific political color?

The Gadsden flag is not always a political statement, but it almost always is. It wasn't until the ~2010s that it became associated with the Tea Party movement. As wiki notes

Starting in the 1970s, the Gadsden Flag became strongly associated with libertarianism. In the 2010s it also became associated with the American Tea Party movement.[22] Because of these multiple uses, the meaning of the Gadsden Flag is ambiguous and depends on the context.[23][24][25]

It's popularity or visibility has subsided some since the early 2010's, but it is still commonly associated with right/libertarian causes.

It is currently used in the banner image for the largest right wing subreddit on reddit:

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I want to offer a counter answer to above for what the flag represents and why it is popular among right leaning groups. (edit: at the time this answer was written, there was only one other answer which is now the lowest rated answer. The following points were written as a response to that.)

• It represents a willingness to re-interpret the Second Amendment, not only the right to bear arms, but also justify modern militias. Nominally this is in opposition to government overreach, but the Gadsden flag is conspicuously absent from protests against Native people's genocide or state sanctioned police violence. This doesn't mean the flag is a symbol of oppression, simply that right wing "government overreach" crowd doesn't align politically with anti-oppression movement. (But the symbol itself does have meaning in itself, separate from political movements. See the evolution into the "Don't Kneel on Us" referencing the Black Panther Party -- though this hasn't seen widespread adoption).

• It represents a willingness to defend First Amendment and freedom of speech in general, as long as that speech favors right-wing politics. (Where are the Gadsden/1A folks defending Bolton's book, or protesting the police violence inflicted upon press (as people protest police violence)? )

• As referenced in other answers, there is some overlap between the political groups that reference the "history" or "heritage" of the Gadsden flag, and what that represents, and other "heritage, not hate" symbols. Nominally, this is in support of the ideal "all men are created equal," but ... Nominally, this is in support of "limited government" in general, but the political groups that associate with the flag are generally missing from left/liberal causes. For instance, missing from protests that convict you of a felony if you protest an oil pipeline on your own land. Or missing from protests about the $25,000 fine a city must pay to remove a confederate statue in Alabama. But just because the flag is missing, doesn't mean the idea the flag represents is exactly opposed.

To summarize: The flag represents a libertarian worldview which is frequently used in support of right wing government, even though some of the underlying libertarian philosophies are shared by those on the left. The context matters, but the flag is usually a symbol showing favoribility towards right leaning views, and this continues to be true at the present time.

Constitutionalism, therefore against liberal-left

The Gadsden flag is an old revolutionary flag, representing a distinctively American idea of personal responsibility and liberty. In a nutshell, the venomous snake on it symbolizes the American people and their readiness to fight any kind of tyranny and imposed will.

In modern times, it has become a rallying flag of various Conservative and Libertarian groups, with a distinct anti liberal-left sentiment. Reasons for that are simple:

• It represents a willingness to preserve the Second Amendment, not only the right to bear arms, but also the right to form militia and implicitly to defend liberty (with arms if necessary) from any kind of government overreach. This goes specifically against numerous attempts by liberal-left through the years to limit gun ownership.

• It represents a willingness to defend First Amendment and freedom of speech in general. This goes especially against the recent spat of people being fired for saying something against BLM, or even if their spouse opposed it. Also against big companies imposing their views of hate speech on internet, censorship etc... Again, this goes against the current leftist narrative of doxxing and harassing everyone who opposes their views.

• Finally, it represents a willingness to preserve the original designs of the Founding Fathers about limited government in general, and demarcation between rights and duties. It goes without saying that the Founding Fathers are not very popular in leftist-liberal circles, with statues of Thomas Jefferson and George Washington being torn down or otherwise vandalized.

Of course, as the Gadsden flag is today seen as the flag of the anti-left (therefore, the flag of the right), it is often mixed with other flags like for example the Confederate flag, although they have little historical connection.

It currently is. From 1 July:

Many held posters or wore T-shirts that said, "Back the blue." ... Two people held up a banner that said, "TRUMP-- KEEP AMERICA GREAT 2020," and someone else stood with a Gadsden flag, which features a coiled rattlesnake and the phrase, "DON'T TREAD ON ME."

https://www.fox13now.com/news/local-news/provo-protest-met-with-counter-protest-downtown-two-days-after-shooting (Note: switched to this article per helpful comment).

For a bigger picture, see this New Yorker article from 2016:

By the nineteen-seventies, it had some popularity in Libertarian circles, as a symbol of ideological enthusiasm for minimal government and the rights of individuals; there was little mainstream interest in the flag as late as the summer of 2001 ... Traffic spiked after the September 11th terrorist attacks, Whitten says, and searches (and sales) also climbed as the Tea Party movement emerged.

https://www.newyorker.com/news/news-desk/the-shifting-symbolism-of-the-gadsden-flag

There's still some uncertainty over what this flag conveys in regards to race, since it's political, and in the US, politics is racially correlated. I think perception of the flag in that manner is currently evolving due to the current widespread racial justice protests. Just this month:

A rally against racism in Craftsbury last week was interrupted by an armed man and two teenagers who drove by the crowd waving Confederate and Don’t Tread on Me flags from the back of a pickup truck.

https://www.sevendaysvt.com/OffMessage/archives/2020/06/16/trio-with-confederate-flag-interrupts-anti-racism-rally-in-craftsbury