Yes, this is exactly what Shopify Markets is built for and you don’t need Markets Pro or any third-party app.
You can assign each domain to its own market.
First add both domains under Settings → Domains, then go to Markets (in your admin sidebar) → click on each market → under Domain / language assign the corresponding domain.
So example.se gets assigned to your Swedish market and example.store to your Italian market. Shopify handles the language, currency, and content switching automatically based on which domain the visitor enters.
You’ll need to translate your content using the Translate & Adapt app (free from Shopify) since the market configuration tells Shopify which language to show, but the actual translated content needs to exist first.
One thing to keep in mind: each domain can only be assigned to one market.
Adding to Mateo’s answer since the domain mapping has a couple of gotchas that bite people right after setup.
Once you assign example.store to the Italian market, double check Italian is actually published as a language for that market (Markets > Italy > Languages). If it is only added but not published, the .store domain loads but still shows Swedish, which looks like the mapping failed when it didn’t.
Second, tune the geolocation recommendation popup (Markets > preferences). It can nudge an Italian visitor who landed on .se toward switching, which fights the clean domain-to-market split you are building.
For SEO, with two separate domains you want each market self-referencing the right hreflang. Shopify handles this once the languages are published per market, but it quietly breaks if a language is only added, not published.
Are the two domains going to be fully separate catalogs and pricing, or the same products with just language and currency differences?
Hello @Nikoscapone
Yes, it is with Shopify Markets. You are able to assign separate domains for each market, so example.se is for the Sweden market, and example.store is for the Italian market.
In Markets bind the two domains and attach each one to a market. From Shopify Translate & Adapt or your preferred translation app, customize the default language, currency, and content for every market.
People who visit a particular domain will automaticaly be directed to the market associated with that domain. Shopify Markets For this, Markets Pro is not needed, and is overkill.