If you derive your sense of worth from a bigger brand and not your own efforts and caliber, you might be complacent for a while then life happens and you will regret not accomplishing anything meaningful in life. Before preparing for your interview and seeking validation for your self worth from big companies, try out internships at startups and get exposed to disruptive ideas. Do not play by somebody's rules. That leads to stagnation. A small plant does not get enough sunlight when it is shadowed by a giant tree. The sunlight you need is exposure to problems you can solve.
Google and Amazon are the new IBM and AOL. They were once the disruptors, now ready to be disrupted by newer generations. Every other day we read about how internet is broken and finance is broken, try starting there.
If you decide to get a job at big company, its not all that bad, just do not take it so seriously.