Zooming out a bit from Winnipeg,
I’ve also seen this kind of tagging in New Brunswick. For example, this intersection in Fredericton has “Rue York Street” intersecting with “Aberdeen Street” on one side and “Rue Aberdeen” on the other. The latter scheme has the visual effect of the rendered map alternating the languages, and I recall also seeing it elsewhere (either somewhere else in New Brunswick, or in Winnipeg?) - though of course it makes quite a mess of routing directions.
I think it’s practical as well as considerate to reflect on why mappers are already doing this kind of tagging in Canada, and what kind of best practices we could suggest that would respect their aims while keeping OSM data useable, maintainable, and as consistent as possible. (Practical because mappers keep on doing it. Considerate given the history and often the official status.)
Certainly we should encourage to have name:en and name:fr tags with appropriate values wherever there appears to be an official or semi-official bilingual name. The name tag for these streets can and should be discussed, possibly regionally.
To give an example of a prominently bilingual region mapped fairly well in OSM: the Brussels standard seems to be “[French name] - [Dutch name]” (e.g. “Rue de la Vierge Noire - Zwarte Lievevrouwstraat” or “Quai de Willebroeck - Willebroekkaai”), which to me doesn’t seem better visually, and doesn’t resolve issues of which language to put in the first position. But I think this illustrates that a tag like name=Avenue Niverville Avenue is not without precedent in OSM. And I’m not sure if anyone here would consider Brussels-style name=Avenue Niverville - Niverville Avenue a better alternative.
The linked thread/post from around 2022 Multiple delimited names in the name tag - #101 by ezekielf suggests semicolon-delimiting the names. This would work well for applications that care to display these names nicely. Unfortunately a lot of default OSM tools - the default carto renderer, the default routers on osm.org - probably won’t implement displaying these names nicely. And while we can and should repeat “don’t tag for the renderer”, it’s unrealistic to stick our heads in the sand and be surprised when some people don’t particularly like their street showing as “rue Percy;Percy Street” when they load up openstreetmap.org. (And I’m not sure it would save us from edit wars about the order.) So we could adopt the semicolon as a solution, but if we do that, we’d probably also want to promote a map viewer (tile server for openstreetmap.ca? apps? etc) that can actually use it well.