From what I understand, the one connected to the Ml isn't activated yet but basically if the princess wants to or has reason to she can turn him into a mindless puppet and if the Oath gets removed an order would be sent to k*ll himself so he would either be turned into a mindless puppet or die. The one on the grey hair girl is basically an oath of silence where she can't reveal anything against the princess. I'm not too sure if there are other oaths and things like that.
𖡼𖤣𖥧𖡼𓋼𖤣𖥧𓋼𓍊𖡼𖤣𖥧𖡼𓋼𖤣𖥧𓋼𓍊𖡼𖤣𖥧𖡼𓋼𖤣𖥧𓋼𓍊𖡼𖤣𖥧𖡼𓋼𖤣𖥧𓋼𓍊𖡼𖤣𖥧𖡼𓋼𖤣𖥧
Spoiler!
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