If I invent something specific then someone patents a similar thing later (let's assume a recent patent e.g. 2015), do I have the right to manufacture and sell products using the invention? And if so, do I technically have copyright but not the patent?
You don't mention the country, but assuming the patent is from the country you are looking to manufacture or sell in then no you can't produce the product without fear of being sued for infringement. It doesn't matter that you believe you invented it first since most all countries have a first to file system for patents. Now, if you publicly disclosed the invention before the patent was filed, you might have grounds for invalidating the patent, but that isn't an easy or cheap process.
I will say that many people confuse patents with applications and it is very possible the document you are worried about isn't an active granted patent. If you were to provide the document number we can advise you as to its status.
Copyright is really a different thing altogether and not really relevant to inventions.
Most likely - No
First, copyrights have nothing to do with it other than any written or drawn plans you might have made can't literally be copied without your OK.
They have a patent and can try to stop you from making something that infringes that patent. An exception would be if you had been already manufacturing it more than one year before the patent’s effective filing date. they applied for their patent. Then you could keep making it, where you were making it, the way you were making it. It's sort of a grandfather right.
Another possibility would be if you had published your invention before they applied for their patent. They still have the patent but you (and anyone else) can use that publication to attack the validity of the patent at the USPTO. If successful, then you can make it. Or you could make it and use the publication as part of your defense if they came after you.
The heart of the patent system is you get a temporary monopoly in return for disclosing your invention to the public and giving the rights to it to the public at the end of the patent term.