Its interesting that you bring up other clubs. I can think of two that both suffered similar fates to Rangers although there will undoubtedly be others.
First example Airdrieonians who when they were liquidated had to reform as Airdrie Utd. They were a new club who were formed from the embers of the old club. This happened mainly because the new owners if I remember correctly could not afford to buy the name Airdrieonians off the creditors. So a new club albeit one that had a connection to the old one same fans etc was formed.
Second example is Fiorentina who went bust in 2002 and came back as Associazione Calcio Fiorentina e Florentia Viola later incorporated into AFC Fiorentina after they bought back the rights to use the name Fiorentina. Notice the could not call themselves Fiorentina out right as this company went bust.
In both cases am sure their fans see themselves as the same club although legally it is a new company.
Now the only difference in Rangers case is because Green and Whyte appear to have had control of the liquidation process they have been able to transfer the name, strip, stadium etc to a new entity in the close season. This has prevented the problems that both Fiorentina and Airdrie faced and helped create the illusion that it was the same club the whole time. If this hadn't happened Scottish football would have had a year or two without a club at Ibrox while the whole process was sorted.
So in short I have no problem with Rangers fans claiming same club etc but legally the club/company are both new entities. You should also probably look at erecting a statue of Green outside Ibrox for saving you.
Appologies for the bold I don't know what happened.