First of all in the Verse you mentioned: أعز and أذل are (plural) nouns in the feminine form أَذِلَّةٍ and أَعِزَّةٍ. And the emphasis is not on those words, so you may have misunderstood or missed the meaning!
While in the dua' they are verbs in the imperative form!
Meanings of أعز and أذل:
أعز: could mean cherish, value greatly or reinforce... All of them and especially the 2nd and 3rd meaning which go ahead with the Verse and the dua'. (For Arabic/Arabic with detailed explanation look here)
أذل: could mean take down, conquer, lower oneself in dignity, discredit ... (For Arabic/Arabic with detailed explanation look here)All of them go ahead with both the Verse and the dua', as i will try to explain
Meaning of the Verse:
You should know that the emphasis in the Verse is not on "أعز and أذل" but on "أعزةٍ على and أذلةٍ على"!
The Verse is talking about people who revert from Islam (and become pagan or kufar etc.). Allah is telling them that if they do so HE is able to bring forth others (whom are described as people) who will love HIM and respect HIM and who will regard themselves as lower in dignity and who degrade themselves towards the believers, reinforce and cherish them, as they consider them as highly respectful people and on the other hand discredit, degrade and humiliate the disbelievers ... (see for example tafsir ibn Kathir). So their love for ALLAH made them harmless, humble and respectful towards the believers but stern, harmful and irreverent towards disbelievers!
Meaning of the dua':
اللهم أعز الإسلام والمسلمين، وأذل الشرك والمشركين
O Allah reinforce Islam and the Muslims (value them highly) and discredit the Shirk and the Mushriks (take them down) ... just to mention one interpretation!
And Allah knows best!