Das ist absolut nicht seine Kragenweite.

Der spielt nicht in ihrer Liga

Die ist ein paar Nummern zu gross für ihn

are commonly use to express "above one's station" (which I haven't really heard for a long time in the UK, to be honest.)

Maybe anmaßen (or related anmaßend/Anmaßung):

Er maßt sich an, mit der Tochter des Direktors anbandeln zu wollen.

Maybe

Ehrlich gesagt macht er sich da falsche Vorstellungen / falsche Hoffnungen.

or

Ich denke, er macht sich da was vor.

or

Er hat, soweit ich sehe, ein paar Grundregeln des Lebens nicht verstanden.

or

... überschätzt seine gesellschaftlichen Möglichkeiten (ganz gewaltig)

or

In meinen Augen lebt er in der Illusion ... (e.g. einer klassenlosen Gesellschaft)

or

Wenn du mich fragst, leidet an Realitätsverlust.

(I'm just trying to narrow down possible translations in order to find one that 'sounds German'.)

It might be expressed as

Er hat Höhenflüge.

However: I never heard someone talk about social positions (in that context). I can't even imagine someone would think about such a thing.

Sich für etwas Besseres halten

like it was suggested in other answers means feeling superior to somebody. That seems to be rather the case for someone who is thinking about social position if it comes to finding someone attractive. So maybe the hospital's director or his daughter

hält sich für etwas Besseres

and therefor it seems unpromising to be interested in her?

Your description of

He's thinking of making advances to the hospital director's daughter. If you ask me, he's got ideas above his station. He's not set up for life or anything!

appears to focus on his position not being sufficiently secured to warrant making undesired moves. There have been several good proposals for a translation of "ideas above his station" already that would fit pretty well with how that phrase would be applied to Flaubert's Madame Bovary.

In connection with the "he's not set up for life of anything" part of it, my first thought here was the somewhat different "Er überreizt sein Blatt.". This is a very German saying referencing the card game "Skat" which starts with a bidding phase based on the kind of game you consider yourself able to win against the two other players. If your announced game does not reach the level of your bid (possibly because you misspeculated about two blind cards you are getting), you lose by default the moment this is discovered (in informal play, you are typically allowed to amend your announcement if no cards have been played yet, but having to play a different game than one prepared for more often than not leads to losing anyway).

So this is a saying for setting oneself up for failure due to unreasonable ambition.

The idiom »have ideas above one's station« is commonly expressed as

sich für etwas Besseres halten

So, to translate your colleague's »If you ask me, he's got ideas above his station« you could have said:

Wenn du mich fragst, hält er sich für etwas Besseres.

Or alternatively:

Wenn du mich fragst, hat er große Rosinen im Kopf.

The meaning of [große] Rosinen im Kopf haben is having high-flying, unrealizable plans or far-fetched, unrealistic ideas.

marry above (one's) station
To marry someone who is of a higher social class or standing than oneself.
A: "I hear that the local fishmonger's daughter is betrothed to a rich foreign lawyer!" B: "My word, she's certainly marrying above her station, isn't she?"
For all the talk that social classes have been wiped away in recent years, you will still find people who believe one can't or shouldn't marry above one's station.

Compared to the question:

He's thinking of making advances to the hospital director's daughter. If you ask me, he's got ideas above his station. He's not set up for life or anything!

He's hoping for (/ He thinks he is qualified for) something unsuitable for someone of his social position.

Answer

That means in German that he has ideas that are nicht standesgemäß

To that dict.cc says:

• normalized adj [conforming to a social norm or standard]


• standardized adj [brought into conformity with a social standard]


• sociol.befitting one's rank adj


• sociol.befitting one's social status adj [postpos.]


• having basic rules or protocols adj


• to marry within one's social class

Making that

Er denkt darüber nach, der Tochter des Krankenhausleiters Avancen zu machen. Wenn du mich fragst, hat er Ideen, die nicht standesgemäß sind. Er ist nicht reich genug oder irgend sowas!

While Stand is not an official characterisation, social position or social status are still distinguished in the Form of Stellung, gesellschaftlicher Rang, Satus and the word standesgemäß escaped the linguistic fogging of implied legal and societal levelling of ranks. As is evidenced by the continued use of the word I suggested:

standesgemäß

Bedeutung

entsprechend den Vorstellungen, die aus der Zugehörigkeit zu einem Stand abgeleitet sind

Beispiele:
eine standesgemäße Erziehung, Heirat, Verbindung

standesgemäß leben, auftreten

sich standesgemäß kleiden
standesgemäß heiraten

Drei behäbige gemütliche Landleute, die eigentlich standesgemäß dritter Klasse fahren sollten

So, since the original describes the situation as 'daughter of director' (desgnating her inherited status) 'advances' (love relationship desired, previously clearly wanted to end in marriage) and 'ideas above his station' (societal status of the Romeo being lower than daughter/director) and 'not set up for life' (Romeo's staus, measured in his financial situation and prospects) any colloquial translation looses precision and switches register. This is about class and a perceived mismatch based on that concept, regardless of what you want the speaker to say instead.

Therefore the original translation offered in the question is in fact already quite good. With a caveat.

Wenn du mich fragst, sollte derjenige sich schleunigst seines Standes besinnen ...

It is too freely transforming the merely descriptive observation of a class-conscious speaker into a prescriptive statement. The prescriptive part is only weakly implied in the original version.

seines Standes besinnen = he should remind himself of his station (and he should stop thinking of daughter)

er hat unstandesgemäße Ideen = He has ideas above his station (while the English is unidirectional aspirational, the German is strictly speaking bidirectional in characterising the mismatch)

Examples abound on that source site:

Der schwedische Geschäftsmann ist ohne edle Abstammung. Heute sind bürgerliche Ehepartner im Königshaus keine Seltenheit mehr. Doch zu dieser Zeit galt eine solche Hochzeit noch als nicht standesgemäß und war demzufolge unerwünscht.