Yes, you can have a legal right to manufacture it - which is to say the legal right to try without being sued or breaking a law. That's not the same thing as the legal right to manufacture it successfully, which is what the distinction would be.
The article is blurring the right to attempt to manufacture without legal restriction, and the ability to successfully replicate the formulation. This blurring makes sense from some points of view (especially from Jacobin's POV, and from an argument of immediate justice for the most people), but obviously falls apart from other POVs.
I think what that person you responded to what getting at is a "country" has the legal right to do whatever they want. If a country signed a treaty promising to recognize patents from foreign countries they have the legal right to withdraw from the treaty.
I assume poorer countries are not doing this due to fear of foreign aid being withdrawn, but this isn't really an issue of "rights".
We're bringing up rights because Jacobin framed in the terms of rights. And within the specific context of the article, we don't need to sorry about worrying about repercussions - Moderna has effectively granted a license to allow 'random countries' to make their vaccine without being suited for patent infringement.
Jacobin's point is that the pledge is meaningless because the patent lacks sufficient information to make the vaccine anyways - it's all locked up in trade secrets. Jacobin then frames this in terms of rights.
Jacobin claims that the way patents are usually discussed will imply that patents are about 'positive rights' - that is, if you have a patent or license, then you have the ability to commercialize (or whatever) the thing that is patented. Jacobin then claims that in reality, patents are just a negative right. Having a patent, or a license to a patent is just a freedom from not being sued for commercializing a product - since the real positive right that people want - the ability to successfully produce the object is actually wrapped under trade-secrets. The article is basically arguing that this split is injust, or at least extremely misleading, and is trying to call out the bullshit.
The article is blurring the right to attempt to manufacture without legal restriction, and the ability to successfully replicate the formulation. This blurring makes sense from some points of view (especially from Jacobin's POV, and from an argument of immediate justice for the most people), but obviously falls apart from other POVs.