Sorry, I may be wrong - I am not a native English speaker, although read and communicate a lot in English. I do not hear often the use of “regime” with “good” countries and leaders - say, Obama regime, Merkel regime.

You’re right. Regime carries a negative connotation and it’s generally why we don’t hear about a Merkel or Obama regime, though technically speaking they are all regimes. Normally we call them “administrations”: Obama administration, Trump administration, etc.

You are completely right. "regime" has clear bad connotations.

Gizmodo is a left-wing outlet so this kind of snark is to be expected.

It's probably because President Obama and Angela Merkel don't/didn't generally do things we associate with regimes. President Obama on numerous occasions at the start of his first term attempted to work cross party. I think it's generally used more when talking about a dictatorial style of government. President Trump, regardless of whether you like him or not, is not a consensus builder. His style is that of directions from the top, no compromises, everyone is either with him or they are potential enemies. I think this is why you would see it used more. I don't know that it demoted a bad or good government more than any other word, nor would I read into it that way. It's quite possible that it was intended to be read exactly the way you're describing.

In Germany, I see “Merkel’s regime” used predominantly by the media that are associated with far-* ideas.

It's definitely a possibility that the intention is there, but I don't know that it's always intentional. I see it far more used by the type of organization than the specific political leaving of that organization. For example in American politics Mitch McConnell is in charge of a portion of the same party at the President. He often agrees with the President and has defended on numerous occasions. Despite this his office isn't generally described as a regime. He governs by consensus. I think that most western cultures tend to view dictatorial, non consensus governments as bad, which means a word associated with these might get a bad rap. In some quick research it appears that your belief that the connotation had shifted appears to be fairly correct. Usage has become more common for heavy handed governments, the word itself doesn't have that as part of the established long running definition however.