No, he was the prime minister, who’s role is to serve as king in his absence.
This was a common role in the ancient far east. The position was called the Grand Vizier in some circles. You’ll see the role first pre-figured in Joseph, son of Jacob/Israel in Egypt. He’s given the Pharaoh’s ring as a sign of his authority.
Eliakim’s role was to manage the household of the king while the king was around, but if the king was off on an expedition, whether it was in battle or visiting another country, he would leave someone in authority to deal with things until he returned.
Does this sound in any way similar to the role of our Pope?