Some time ago, I flagged something on meta as "rude or abusive". Checking now to see what became of it, I see my flag "aged away". I understand that close votes can age away, but I didn't know that "rude or abusive" flags could age away. Is this a way for the moderators to decline a flag without declining it?
As explained by Martin such flags age away after 2 and 4 days for posts and comments, respectively.
"Rude or abusive" and "spam" are not flags that are intended to always (or even usually) involve a moderator. Instead, once enough flags are accumulated the content gets deleted automatically. I think the idea is roughly that the main use-case of those flags is blatantly inappropriate stuff, so that if there is no quick accumulation of flags, then it likely does not fall in that category (while it could still be inappropriate content). However, especially for (late) comments there can be the issue of just not getting enough visibility.
Yet, moderators also get shown those flags and can and do react on them.
On your question:
Is this a way for the moderators to decline a flag without declining it?
When enough flags accumulate the action will happen, no moderator is needed. In that sense, if a moderator is convinced such a flag should be declined, then they better actually decline it rather than to try to sit it out.
I doubt moderators intentionally let such a flag unhandled to let it age away. What I think could happen is something 'in the middle,' that is, a first mod sees the flag and is not quite sure about it. Thus, they let it around for the next one to have a second opinion, etc.
If you are still convinced the content should be deleted, I would flag it anew. For this you might use an "other" flag and explain the situation. Note that moderators can give a reason for their action on a flag when the flag is on a post (yet not a comment); thus in complicated situations you might decide to flag the post even if it is about a comment to allow for this. (Of course this is meant for the "other" flag not "rude or abusive.")