You can use ITaxonomyRetriever to retrieve the taxonomy and its tags which will include all of the tag fields -
// An instance of ITaxonomyRetriever (e.g., obtained via dependency injection)
private readonly ITaxonomyRetriever taxonomyRetriever;
// Retrieves the general TaxonomyData object for a selected taxonomy
TaxonomyData taxonomyData = await taxonomyRetriever
.RetrieveTaxonomy(CategoryName, "en");
// Retrieves the tag by Identifire
Tag tag = taxonomyData.Tags.Where(x => x.Identifier = tag.Identifier).FirstOrDefault();
string tagDisplayName = tag.Title;
string tagCodeName = tag.Name;