Specific problem happening with the font Inter in Illustrator. We can see that it's used but when we select the text box, we can't see the name of the font.

A number of different technical issues can be happening at the same time with freely available fonts such as Inter.

First of all, IMHO, Google really needs to work on quality control of fonts hosted at the Google Fonts web site. Over the past several or more years I’ve encountered odd technical problems with a variety of different type families downloaded from the site. Occasionally I can go back to the site months later, download a certain type family again and find that version working properly. It’s obvious at least some of the fonts there receive updates and bug fixes from time to time. I just wish they did better up front testing on the fonts before uploading them live on the Google Fonts site.

Another issue: type families like Inter can be downloaded from more than one place. Google Fonts has it. But so does the Font Squirrel web site and some other free fonts web sites. Not all of those sites has the same build of those font files posted. That difference alone can cause conflicts. If multiple people are sharing and editing documents using Inter every person needs to be using the same exact font files. That’s one advantage of syncing fonts from the Adobe Fonts site: everyone would be using the same file version.

Font management apps can cause their own problems. A buggy font file may act just fine in Adobe Illustrator yet immediately crash a rival vector app like CorelDRAW when the font is selected in the menu (Barlow Variable is one example that does just that). Windows doesn’t always do a good job of removing font data from various locations in the OS when a font is uninstalled from Windows’ Fonts folder. That’s one possible way how switching back and forth between static and variable versions of a certain font won’t solve a problem with fonts that are misbehaving.