In its broad historical strokes, yes, Roman history looked pretty much the same in the Middle Ages as it does today. Nor should this surprise us, since our story of Roman history is based fundamentally on the narrative found in the Roman sources themselves and Medieval people had access to those Roman sources too. There are of course some differences, notably neither Livy nor Tacitus were widely read in the Middle Ages, but a course grained history of Rome doesn't in any way depend on reading either Livy or Tacitus. Throughout most of the Middle Ages, there were two major, widely read histories of Rome: Eutropius (plus his continuers Paul the Deacon and Ladolf Sagax) and Orosius. Both of these provide a narrative of the Kingdom from Romulus to Superbus and the expelling of the kings and the rule of the Consuls (Eutropius, 1.1-9; Orosius 2.4-5). Both likewise discuss the punic wars (Eutropius, bks 2-4; Orosius, bk. 4). These could of course be supplemented by information found in other widely read Latin classics like Lucan and Sallust, but those form the fundamental outline that would have been familiar to most educated authors.

We can see how this looked in practice in the highly influential twelfth century history by Hugh of Saint Victor. It is a fairly encyclopedic history, that is comprised mostly of lists of names and regnal years, with the occasional comment to contextualise what is going on.

You can follow this very pretty later 12th century manuscript from Saint Victor, whose folio references I'll be giving, though I'm taking the text from Mortenson's edition.

We start on 7r at the bottom of the right column with the "Regnum Latii" (Kingdom of the Latium):

The Kingdom of Latium which arose under Ianus, as they call their first king, ran from the 76th year of Ehud judge of Israel, until the 7th year of Darius king of the Persians. That is 80 years after the Babylon exile, 10 years after the return of Babylon and 4 years after the construction of the temple. It lasted 878 years, through 6 Latin kings, 14 Alban kings and 7 Roman kings up to Tarquinius Superbus, as the subject matter teaches.

Regnum Latii sub Iano ut dicunt primo rege exortum ab anno LXXVI Aioth iudicis Israhel usque in annum VIII Darii regis Persarum, qui est LXXX post transmigrationem Babylonis, X post reuersionem Babilonis, IIII post instaurationem templi, cucurrit, per annos DCCCLXXVIIII et reges Latinos VI, Albanos XIIII, Romanos VII usque ad Tarquinium Superbum, ut subiecta docent.

Then on the fourth column of 7v we find a list of these kings, first of the Latins (Latinorum) then the Albans (Albani) then the Romans (Romani). After which we have another short description, this time of the expulsion of the kings:

After this the kings were expelled from the city and the consuls managed the republic until Julius Caesar, that is up to the 20th year of Hyrcanus, son of Alexander, king and priest of the Jews, through 462 years.

Post hoc, expulsis ab urbe regibus, consules rem publicam tractabant usque ad Iulium Cesarem, hoc est usque ad annum XX Hyrcani regis et sacerdotis Iudeorum filii Alexandri, per annos CCCCLXII.

Then on the third and fourth column of 8r we find the list of consuls and the year of their rule counting from 1 up to 486 with Caesar Augustus. Here we have another bit of context:

We have selection these from the period of the consuls (from when the kings were expelled up to Julius Caesar), recording individually their years and the time at which they managed the republic after the kings, and have marked down those especially during whose time particular famous events done either by them or by others are recorded, for example, at the time of consul Papirius the things that are written about Alexander the great, at the time of the Scipios the punic wars, at the time of Marius the Jugurthine war and at the time of Cicero the Catiline conspiracy.

Istos de temporibus consulum excerpsimus ex eo quo reges urbe pulsi sunt usque ad Iulium Cesarem, singulis annos suos et tempora quibus post reges rem publicam tractabant adnotantes, et eos magis signauimus quorum temporibus uel ab ipsis uel ab aliis aliqua insignia gesta referuntur, ut uerbi gratia temporibus Papirii consulis ea que de Alexandro magno scripta sunt, temporibus Scipionum bella Punica, temporibus Marii Iugurtinum, temporibus Ciceronis Catilinarium.

At this point we must leap forwards to 24r for the Roman empire. I skip the first paragraph where he explains that he will try to follow the order of kings and kingdoms from the time of the incarnation, and explaining the division of the lists prior to the incarnation from those after the incarnation. Noting also that there are difficulties in the computation of dates that the prudent reader may need to sort out.

After the time of the consuls, monarchs acquired imperial power in the Roman republic. Julius Caesar seized imperial rule first, and the custom of Rome being subject to one ruler returned – in the year 722 from the founding of the city, 630 from the expulsion of the kings – called now emperor instead of king or a more sacred name after Augustus.

Post tempora consulum monarchi in Romana re publica imperium obtinuerunt. Primo Iulio Cesare arripiente imperium, repetitusque est mos Rome uni parendi, anno ab urbe condita DCXXII ab exactis uero regibus DLXXX pro rege imperatore uel sacratiori nomine post Augusto appellato.

I also leave out the 3rd paragraph which is just explaining that since Jesus was born 42 years into Augustus's reign, the count of the incarnation on the right begins at 14, which is the remainder of Augustus's rule after the birth of Jesus.

Of course, as we saw from the comment on the consuls, Hugh is expecting the reader to be able to fill in the notable events from their reading not only of Eutropius (or whichever continuer) but from their reading of standard school texts like Sallust (Jugurtha and Catilina), the histories of Alexander (e.g. Justin), Lucan, etc.

So in summary, while the sort of things that a medieval reader might have been interested in about Rome would have been very different from the modern day, the outlines of Roman history itself are not radically different. What differences there are are going to be found more at the level of details about specific events, rather than at the level of overarching narrative.

Edit: added a brief conclusion and touched up my fairly rough translation in a couple places