Nobel Prize in Physics 2019 on exoplanets and Hot Jupiters The 51 Pegasi b, discovered by Didier Queloz and Michel Mayor  is classified as a hot Jupiter and not considered to be a star. I had read a bit about the minimum mass necessary for being a brown dwarf at some 13 Jovian masses, also the following question "Can Jupiter be ignited?" which has an answer stating that clear boundaries have not been drawn regarding gas giants and brown dwarfs. My question is how was it concluded that a Hot Jupiter different is from star types like brown dwarfs, in the sense that they could assert it wasn't a star they were observing ( keeping in mind the possibility of more massive ones like WASP-18-B (around 10 Jovian masses)).
Specifically how does one eliminate the possibility that it could have been a star at the end of its life-cycle for we could have considered it to be a smaller star of a binary system (considering the case of distance to be pretty close to the central star) ? 
Apologies if the question sounds too naive.  
 A: Your question is not naive at all, don't worry.  
51 Peg b has been found via the radial velocity method. Upon detection, this method automatically gives a minimum mass of the companion, which is $M_{\rm min} = M \; \sin(i)$, where $M$ is the unknown actual planetary mass and $i$ is the (generally) unknown inclination of the orbit towards the observer.  
It was determined that $M_{\rm min}\approx 0.5 M_{\rm Jup}$, and for isotropically distributed orientations of orbits on the sky, the expectation value of $\sin(i)$ is about unity. Extremal values of $i$ that would make $\sin(i) \ll 1$ and thus $M \gg M_{\rm Jup}$, are improbable, but not excluded. In their paper, Mayor & Queloz cite a $1\%$ chance that the planet is above $4 M_{\rm Jup}$ and $1/40.000$ that it is above the deuterium-burning limit of $13 M_{\rm Jup}$.
Thus it was concluded that an object of about Jupiter-mass was found.
That this object is also a planet stems from the difficulty of making a stellar companion (typical binary stars have about equal mass, but ratios of up to a factor of 10 are normal), and then evaporating it enough to loose not 100% of the mass, but just about 99%, which is another improbable mechanism.
Considering the last part of your question, stars don't just become 'small' at the end of their lives. Stars with significant mass-loss at the end of their lives loose this mass via strong stellar winds, which a) leave very visible planetary nebulae (which is absent in 51 Peg) and again b) leaving such a small blob of mass (compared to a hypothetical progenitor star) of a tiny fraction of the original mass is improbable, or even impossible (strong winds cease to exist at certain stellar masses/radii).
