Yes, that island to the northeast of Ruathym is almost certainly Axard, although proving it beyond a shadow of a doubt is difficult.
The article in Dragon Magazine
Perhaps definitive?
Quoting from the article "The Silent Shadow: Raulothim" in Dragon Magazine #253, by Ed Greenwood:
[Raulothim] spends long days lying motionless, gazing out over the North from the lip of his lair in the Pit of Stars, the cauldron of an extinct volcano (known to the Netherese as Xardmount) on the rocky island of Axard, the north-easternmost isle of Ruathym.
and:
The events that made Raulothim so thoughtful began with his chance encounter with a ship that sailed the skies. The ship's spelljamming crew were seeking to hide treasure on the "uninhabited" outer islets of Ruathym.
This confirms Ruathym is made up of more than one island, the north-easternmost is named Axard, and that there are outer "islets" of Ruathym, and furthermore the quotes around "uninhabited" suggest that common belief is that these outer islets are thought to be uninhabited, but they are not, since one is inhabited by Raulothim.
This all but nails that the island you are referring to is indeed Axard. Alternatively, there could conceivably be an uncharted island in that general vicinity that is Axard. That would also fit the evidence I've cited. But it seems unlikely.
5th Edition Materials
In addition to Fizban's Treasury of Dragons, Ruathym is mentioned in the Sword Coast Adventurer's Guide and Storm King's Thunder. No answer in either place.
Sword Coast Adventurers Guide, 5th Edition
Ryathym is mentioned, but the island is not named; there's a map, no label on the island.
Storm King's Thunder, 5th Edition
Ryathym is mentioned, but the island is not named; there's a map, no label on the island.
Forgotten Realms Campaign Setting, 3rd Edition
Contains a moderately detailed section on Ryathym, the island is not mentioned, on maps it is unlabeled.
"A New Map of Faerun"
A really comprehensive and impressive fan-created map by Adam Whitehead shows the island as Axgard, and shows a feature on the island as Xardmount (Pit of Stars). While fan-created, the author reports relying on source material. Perhaps further investigation of the author's sources would show the original map with the island labelled.
Enquired in the comment section of the site, the author confirmed that an official source (the Forgotten Realms Interactive Atlas) reports that island as Axard:
Yes, that comes from the Forgotten Realms Interactive Atlas (1998), which lists the island as such. The original source was Ed Greenwood in a Dragon magazine article (#253) about the dragon Raulothim, who lairs on Axgard.
The user Roflo confirms it in the comments to this answer (see a screenshot here).
Forgotten Realms Wiki
The Forgotten Realms wiki entry on Raulothim cites the Dragon article and largely echoes details from that article. The list of references at the end of the wiki possibly contains a reference that would have a labeled map, but unless it were a reference from Ed himself it's hard to see how it would be more definitive than the Dragon article.
Candlekeep Forum
There are other hints in the Candlekeep Forum. Sifting through this and sorting out official references from fan-made references is...daunting, but there may be a link to a reference to a product that contains a definitive answer.
The "Questions for Ed Greenwood (2009)" topic
One such is this topic here at forum.candlekeep.com, search the page for both "Axard" and "Axgard".
The "All that is Ruathym" topic
This topic on Candlekeep contains a link to an unattributed map that labels the island as Axard.
Some other random map
Finally, there's a map at realms.dndpedia.com. Warning, the site does not use https, and while it seems safe enough, that's a bummer, so I only linked the jpg. The map seems to depict that island as labeled "Axgard", and although I can't tell the provenance of that map, it does not appear totally fan-created. The site contained no attribution. Dan O'Shea found map art on Mark Taylor's deviantart, where there are a number of Forgotten Realms maps. Hard to say the relationship to "official". Dan also says the island to the North of Ruathym is labelled "Axgard" on his other maps of Faerun too.
Conclusion
Beyond the Dragon article, if there's a "real" answer, it is buried in the sands of time, perhaps in one of the references listed in the Forgotten Realms wiki article. However, even if some book, even officially published, were to name that island Axard or anything else, how trustworthy would the answer be? Much like the real world, the Forgotten Realms is layer on layer of various people naming and renaming things, telling and retelling stories. Even Ed himself may have named the island different things at different times.
So I'm going with: Yes, that's Axard.
