Fundamentally, You could argue that he did, but probably not in the way the question implies. Mitsuki is a clone, so in that regards he is more or less human. His emotions are going to be derived from Orichimaru's personality, especially his younger self, and how he was raised. In that regard, Orochimaru did do the activity to make him think for himself, erasing his memory on failure. As a side effect, Mitsuki's emotions are fairly low in intensity.
Orochimaru then used the success of that activity to point Mitsuki to Boruto. With that, Mitsuki was nearly blank shell emotion wise, as shown by his general lack of strong emotions. He was only allowed to leave once he matured enough to think for himself, and then he was sent to Boruto. Most likely, that is the only real "designed" part of Mitsuki's emotions, that Orochimaru forced him to mature and then gave him a "sun" as they are so called in the Naruto universe. Mitsuki in turn becoming friendly to Boruto is his own thing.
It could be the case though that this all came about because of unseen manipulation by Orochimaru. Mitsuki is a clone, yet is also a lot more kindhearted than Orochimaru, which could mean there is some manipulation to tone down that apathetic/evil nature. It could also be though that was more Nuture than Nature, so In truth there is no full and clear answer, in the Manga, Anime, or any other source. The Anime will probably give some answer in a few episodes though, as its fairly clear Mitsuki wants to answer that very question himself before he returns to the Leaf. This is Anime only though as far as I know.
There is some manipulation as shown by the experiment Orochimaru did, but that was one to teach Mitsuki to think for himself. There is also the fact that he was pointed to Boruto, but that is more priming than design. Mitsuki is mostly a clone of Orochimaru, so he is at least mostly human and has his own emotions, so it was still mostly his own choice to be friends. That leaves one unknown, if he was engineered/designed to be like he is emotionally or not, which could count, but is unknown. Regardless, Mitsuki was free to make his own decisions after he passed the test, and he chose to checkout this person whos picture he was given (Boruto). He wasn't forced, tricked, or deceived into anything beyond meeting him.