Is getting an oral recommendation from a professor who has known me since childhood a conflict of interest during graduate admissions?

First, I don't think you did anything wrong. If there was improper action anywhere it was on the part of others. (See the final statements about "favors" in the answer of tbrookside for example.) But I suspect that the oral recommendation given had little weight other than at the margin. The professor wouldn't have taken you on if there was any doubt about your abilities. It would have been foolish to take someone unqualified as a "favor" to another.

The other student also has no reason to be jealous as they also got accepted.

But the social thing is something you will have to work through. Demonstrate your competence in the usual way and it should tone down over time.

I suspect @Buffy is right, that you did not do anything wrong. However, you did receive an advantage you did not earn because of who your parents are and who do they know. I suspect that that advantage is much more subtle and much more substantive than the oral recommendations (compare your experience with this answer Does politics and perception play a role in higher education? )

In short, you were lucky, and it is not moraly wrong to be lucky. However, it is understandable (not right, but understandable) that people who had to earn their position the hard way require extra evidence before they are willing to accept you as an equal. It is your challenge to deal with this situation gracefully. Demonstrate your competence, as Buffy suggested, but also learn how priviledged you are, and how challenging university can be for those who do not come from a middle class or higher background. A nice place to start is here: https://www.asanet.org/sites/default/files/attach/journals/feb15asrfeature_0.pdf

The purpose of a letter of recommendation is to supply information about the abilities of an applicant to decision makers. The letters are only necessary because those decision makers don't possess direct knowledge about the applicant. When making a hiring or admission decision, I would have no need to receive a letter of recommendation about an applicant I already knew very well; the letter would add no information to the process that I didn't already possess.

If you already had supplied letters of recommendation from other professors, an additional oral recommendation only makes more information available to the decision makers and therefore cannot be harmful or "unfair".

This assumes, of course, that Professor A supplied an actual recommendation. It's one thing if Professor A said, "I have known this person for 20 years and they are incredibly bright and diligent," and quite another thing if Professor A said, "This applicant's father is an old friend of mine and I will owe you a favor if you admit them."