Yes, financial engineering is a part of economics. Financial engineers are continuously working to crunch out less risky, more profitable financial instruments. One example of financial engineering being intertwined with the economy has to do with the mortgage-backed securities that were traded in record high numbers prior to the 2008 crisis.
Bankers were able to take large numbers of warehoused mortgages off their books by pooling the mortgages together, dividing them up into individual securities with small fractions of many mortgages in each one, and selling the contractual rights to that income as a derivative. This is a problem when ratings agencies are stamping these as investment grade securities when they're really junk.
Also, swaps are a clever instrument devised by financial engineers. They can range from simple to complex, and there are many variations of the "swap". I won't get into all of them, but they also played an important role in the financial collapse of '08. Since so many of these financial instruments are traded at enormous volumes on a daily basis, most of which were created by bankers (who call themselves financial engineers), I would have to make the argument that financial engineering is involved in economics. The argument could be made that our economy is actually predicated upon such financial instruments.
And for the other guy, an MBA would be more prudent than the mathematical finance degree. Furthermore, mathematical finance and financial engineering aren't the same thing. One is a legitimate discipline, and the other is a phantom discipline, respectively.