I think Solas' mug is like Sera's bangs, or the hearts on Cassandra's armor- bits of characterization integrated into their appearance/aesthetic. I don't imagine Sera talks about her hair at all, but the way she hacks it off as needed offers insight into her personality. Solas' mug on Solas' backpack tells us that he's a expert wanderer, someone who has streamlined their life to the point where it can be folded up and arranged and carried on their back.
Though I agree this is more likely ...
If we're going down the rabbit hole we might as well make it to Wonderland.
Will he share the mug?
And though I wouldn't turn away to offer.
I would rather he be possessive over his mug, if it is brought up in conversation.
Edit: Spoilered gif that turned huge for some reason.
Something his that he can carry with him, rather than just a tool to be replaced with wear.