Well, there’s not the slightest difference between the two forms of {hamartia} there (both are {hamartian}, nouns in female accusative singular form).
I’m not convinced there’s anything in the immediate (maybe also local) context of Rom 8:3 that requires {hamartias} (a genitive form to match the genitive preposition {peri}) to mean a sin-offering, and again it doesn’t have the slightest difference in form from the acknowledged phrase “in likeness of flesh of sin” (a genitive prepositional form again for “of something”) immediately before it, nor immediately after it with “[God] condemned sin” except for the form being accusative to match being a direct object of the verb.
On the other hand, the context for Hebrews 10:6 definitely involves an implied sacrifice for sin with the same prepositional phrase as at Rom 8:3, {peri hamartias} “about/concerning sin”. (Peri when used in genitive form like this is a metaphorical application of its usual meaning, found in the accusative form, of “around”, as we might say when talking about a topic the topic is fully “covered”.)
But then that particular phrase isn’t being used at 2 Cor 5:21. The term is simply a direct object of the verb, “[God] made sin”.
On the other hand again, Gill is correct that Psalms 40:6 involves a sin offering using a rare form of the Hebrew term for “sin” which in all other occurrences does mean “sin”. Or once (with a slightly different attachment) “become sin”! – but then a sin offering representatively does become sin in the ritual, despite not being a sinner itself. In fact, that particular verse and its contexts are generally regarded (by Christians at least) as Messianic prophecy: God didn’t want sin offerings and sacrifices, but prepared a body for the singer to come and do the will of God in righteousness and to announce good news and righteousness in the assembly. (This is in fact what the Hebraist is quoting at 10:6.)
Then again again again, in the three Greek versions of that verse I can find offhand (the Apostolic Bible, the Majority Orthodox text, and the Jewish LXX – the EOx M text numerates the verse at 41:7 instead), the term is part of the same prepositional phrase found in Hebrews 10:6 (as might be expected), {peri hamartias}.
So eh, nothing in the language seems to decisively settle the matter so far; but in the undisputed examples the sin-offering, when mentioned in Greek, is described with the prepositional phrase {peri hamartias} absent from 2 Cor 5:21. So that does lend weight in the direction of not meaning sin-offering.
But then again, the context of Hebrews 10 explicitly indicates (for example at verse 12) that Christ was offering Himself as a sacrifice for sins, therefore as a sin-offering sacrifice (even though He isn’t called a sin-offering there). And later at verse 18, when talking about the sin-offering specifically, the same prep phrase comes up {peri hamartias}.
So on one hand, it is indisputably canonical that Christ is being offered up as the sin offering (and also the Passover offering which isn’t quite the same thing, one being burnt outside the city and one being cooked and entirely eaten at a meal), an offering that is itself not sinful but which comes to represent sin being put away and destroyed so that people may become righteous to God; and on the other hand, the terminology normally associated with the sin offering is missing at 2 Cor 5:21.
Beyond all that {inhaaaaale!} 
, Hebrews 10 continues a line of thought from Heb 9, discussing why Christ is superior to high priests, and a big part of the Hebraist’s argument is similar to an argument made by Paul in some other places (notably in Galatians): Jesus sacrifices Himself as a descendant of Abraham to keep up Abraham’s side of the Abrahamic covenant, for the sake of all descendants of Abraham (which are all rational creatures since God Incarnated as a descendant of Abraham) who have rebelled and thus broken the covenant which Abraham intended to make but which God graciously spared him from actually going through with, meaning the Father and the Son made the covenant between themselves. That means the covenant (of the promise, unlike the Mosaic covenant) cannot be broken by anyone’s sin (because Abraham didn’t actually participate, so the covenant was only about Abraham and his descendants, between the Father and the Son); it can only be broken by either the Son or the Father, neither of Whom are ever going to break covenant with each other. But because the Son stood in as a descendant of Abraham for Abraham and all of Abraham’s descendants (i.e. every created person, numbering as many as the stars in the sky or grains of sand at the sea, poetically speaking), if any person sins then the Son is the one who pays for that sin, requiring the death of the Son despite Him being sinless Himself (because that was the type of covenant made, passing between hewn animals to show that if either party breaks the covenant they’ll be slain like the animals).
But since the Son is sinless Himself, the point of dying wasn’t to satiate the wrath of God, but rather to keep the covenant: the covenant between Father and Son only breaks if the Son refuses to sacrifice Himself for the sins of other people. And the promise given was that all Abraham’s descendants would be led finally into righteousness.
That means the passion on the cross, among other things, is an enacted assurance that God intends to fulfill His promise to Abraham and to Abraham’s descendants (who are all persons created by God, thanks to the Son Incarnating as a descendant of Abraham): the promise being to reconcile all things to God which need reconciliation, whatever those things are, whether things in the heavens (i.e. rebel angels) or things on the earth (rebel humans for example) or things under the earth (currently dead humans and even slain rebel angels). And if we have been reconciled to God through the death of His Son (as Paul says in Romans 5), how much moreso shall we be made alive into His life! (i.e. there is no reconciliation that ends with permanent sinners, or with annihilated sinners, or with people no longer sinning but somehow not sharing in God’s own eonian life.)
So the topic does have connection to Christian universalism. 
(There’s another big connection in the contexts of 2 Cor 5:21, too, including very interesting contexts in the chapter of Isaiah cited by Paul there, pointing toward post-mortem salvation! – and which has direct connections to the kings of the earth entering the NJ in Rev 21!)
