Just in case it hasn't been very clear, no and its not even close (not that close would do you much good, either). They aren't even close cousins.
ARM's main market focus on 200MHz or lower application, while MIPS focus on high frequency application
I'm not sure that knowing the 'market focus' helps in any way. There are arm variants and some go beyond 600-800 MHz of the top end mobile 'phone processors and there are multi core variants. There are also very popular 50 100 MHz for deeply embedded applications. About the only thing that ARM processor applications have in common is a low power consumption per mips (not the MIPS processor type of mips).
Equally, there are many MIPS processors, and if you consider the Microchip processors there is an application area overlap with, say, the Luminary Coretex arm processors...but just because there is an application area overlap doesn't make the instruction sets compatible, in any way. And without instruction set compatibility, you can't usefully execute code for one one the other.
Historically, it is true that the MIPS processors have been more used in external memory/expansion bus applications where the ARM chips have been more used in true microcontroller applications, but that says nothing about the instruction architecture, and that is what is important here.
(There are, of course, higher end MIPS processors as well.)
I believe there is protected intellectual property in each architecture (partially, in the case of MIPS, see 'longsoon' for example) and they are not something that someone else could just adopt without a license agreement or a legal battle.