Reflecting on the whole "looking for a dream" in Berserk I concluded that Guts' journey for a dream of his own was not really about a dream, but more so about finding choice and agency.
To explain my thought process: from the get go, he's not really given a choice at all. Guts never chooses to be a mercenary; this is something the circumstance sprung on him. He embraces it because it's what was offered and gives him a means to survive.
However, I think up until the point where he accidentally eavesdrop on Griffith's speech to Princess Charlotte it didn't really occur to him that he could have a choice. Guts had been going with the flow, taking orders, doing what needed to be done so that other people's goals were achieved. I don't think he consciously imagined he had no choice; more so, I think it never occurred to him that he was allowed to want something, or to find personal satisfaction for himself.
The moment when that happens gives an interesting context to that: he has just carried out an assassination for Griffith, but did he really believe in what he was doing, or was doing that just because it was an order and that's what he was trained from a very tender age to do?
After leaving the Band of the Hawk, we see that he has found that agency. He realizes he can choose what he'll use his sword for and pick his own battles. He's not just going with the flow anymore. He will now choose to fight for the causes he believes in. To me, this is why he says his dream is not a concrete destination; this is a lot more about having a choice than it is about reaching an objective.
Unfortunately we don't get to see Guts' journey after finding his something, because the eclipse messes up everything, but I'd say we have pretty good evidence in canon that it would have looked very different from his original path. He's now more selective about what battles he fights, because he has found the power to choose. Asking Casca to join him is a great conclusion to this "dream arc": prior to this journey of discovery, I imagine he was a lot less likely to ask for things he wanted or needed than after. And at that point, it's clear he wanted to stay close to Casca, while still pursuing this newfound ambition.
