It means that because of his age, people would consider that he was too old to perform within his profession, and so he would be passed over for promotion, not given new responsibilities, etc. . (for example, in the tech industry, 40 is considered ancient)

Yes, you are right. The author is talking about aging.

The expression "professional oblivion" in this sentence means that it would be the end of his or her career, or that it would be harmful to his or her career. It's likely a purposeful exaggeration and the writer probably does not mean it so literally. The meaning is that the profession he or she is in favors younger people over older people.