Interesting question. Thanks for posting!
When I think of “Christian” omens, I think of some bible prophesy, for one. We were given signs to be aware of the second coming. But when we constantly focus on certain events as “proof” of Christs return at any given period, I think we become idolators. We make ourselves God, the Father, because only He knows the hour and minute of Christ’s return. Much of the “prophesy” in the book of Revelation, from a Catholic perspective , includes events that already occurred or refers to eternal realities in heaven. So, identifying bible “prophesies” creates a type of omen. I dont think that is necessarily what is mentioned in the Didache, but I do think it is something that we should be concerned about not doing. I would imagine the more relevant meaning was to not look at certain signs to guidepost your life. But, if you are making decisions on your life ( eg. quitting your job to evangelize full time, not working to improve your community, or society in general, stockpiling resources) according to “bible prophesy” , other than continuing to live in Christ , then I think this would an applicable meaning as well.
As far as the authority of the Didache, for the Catholic Church, I believe it is considered as having some authority in terms of the Tradition of the Church that comes from the Apostles. Its not biblical authority but it is the Tradition of the Apostles that the bible refers to when it says “hold fast to the Traditions of Apostles, both written and unwritten”. This means it is completely consistent with Catholic teaching.