I honestly don't see how Robin encourages projection. He/She seems like too much of his/her own character for that.
By RPG (and especially Bioware RPG) standards, I'd agree- so long as you think projection requires role playing. By role of audience surrogate, I'd disagree.
Robin has a couple character traits typically used in fiction (and especially in Japanese media) to encourage viewer identification. Customization is a significant one, second only to the classic amnesia trope. Robin, who is also called the Avatar (a double meaning in its own right), is the introductory device used to introduce the players to things they wouldn't/don't know. For most of the first arc, Robin's characterization is pretty minimal and open-ended: the Plegia arc is about Chrom and his development, while Robin is a supporting character whose role and characterization is mostly just being a decent, ordinary person (with a head for tactics). Robin's personality is at its most 'normal', and the easiest for people to identify with if they were in those circumstances. For most of Act 1, Robin is less 'the amazing person doing amazing things that shapes the plot' and more 'the person who doesn't know what's going on being caught in the flow.' Which is basically what the new player always is.
It isn't until the Valm act that Robin starts to really express characterization in his/her own right- actively shaping events, rather than passively responding to them or following Chrom's lead. It's part of the Avatar's rise into the role of deuteragonist, equal to Chrom, with the narrative focus switching from Chrom's development arc to the Avatar's towards the end. (Unsurprisingly, the Valm arc is also the part that, mechanically, most Avatar support pairings will normally start maxing out and children start getting recruited- the support conversation base with the strongest characterizations.)