I think it’s safe to call the HRE an empire, because it did control a comparatively large number of cultures, peoples, nation-states, etc. Certainly Great Britain had an extraordinary reach around the globe–the sun never sets, right?–but it would be appropriate to refer to the HRE as an E.
Holy and Roman, IIRC, had as much to do with politics as it did with religion. Probably more so.
Some of the emperors we refer to as “Roman” really weren’t Roman either. Here’s a list from one site.
Augustus - Roman Italian (born at Velitrae, southeast of Rome)
Tiberius - Roman Italian
Gaius “Caligula” - Roman Italian (born at Antium)
Claudius - Roman Italian (born at Lugdunum in Gaul)
Nero - Roman Italian (born at Antium)
Galba - Roman Italian
Otho - Etruscan
Vitellius - unknown, presumably Italian
Vespasian - Sabine
Titus - Sabine
Domitian - Sabine
Nerva - Italian or Cisalpine Gaulish (born at Narnia)
Trajan - Umbrian Italian (family had settled in Spanish Baetica)
Hadrian - Picenine Italian (family had settled in Spanish Baetica)
Antoninus Pius - Gaulish (born in Latium)
Marcus Aurelius - Gaulish and Spanish
Commodus - Gaulish and Spanish (born in Rome)
Pertinax - unknown (family of slave origins)
Julianus - unknown, probably Roman Italian
Albinus - Italian
Pescennius Niger - Roman Italian
Septimius Severus - Carthaginian and Celtic
Caracalla - Carthaginian, Celtic, and Syrian
Macrinus - Italian (family had settled in Punic Africa)
Elagabalus - Syrian
Severus Alexander - Syrian
Maximinus Thrax - probably Dacian (supposedly Thracian and Sarmatian)
The Gordiani - unknown (probably mixed Italian, African, and Asian)
Philip the Arab - Mesopotamian or Syrian Arab
Decius - Illyrian or Pannonian (born at Budalia near Sirmium)
Trebonianus Gallus - Perusian Etruscan
Aemilian - Mauretanian
Valerian - unknown
Gallienus - unknown
Claudius Gothicus - Illyrian
Aurelian - Moesian
Tacitus and Florian - unknown, possibly Italian or Danubian
Probus - Illyrian or Pannonian (born at Sirmium)
Carus - probably Gaulish and Greek (born at Narbo)
Carinus and Numerian - probably Gaulish and Greek
Diocletian - Dalmatian Illyrian
Maximian - probably Illyrian or Pannonian
Carausius - Menapian Gaul
Constantius Chlorus - Moesian or Dacian
Galerius - Dacian
Severus II - Danubian
Maxentius - Danubian and Syrian
Constantine I and his sons - Moesian or Dacian