Now, I don't think your question meant to say "do these phrases mean raise/bring up in the same sense as parents raise a child? This seems to be what the other answers (and all the downvotes) are concerned with. And as far as they go, they are correct: neither in common usage nor in their convoluted history are those senses ever the same; even if growing up as an orphan in a university, you would be raised in it, and not by it.
But you asked in a very precise way if the verbs can take university as a subject; this implies to me that you were asking if these phrases had other meanings, perhaps related to the etymology of why they are applied to child-rearing. If you don't mind an answer that is obscure and not constrained by contemporary usage, then two of the three phrasings appear to be perfectly acceptable and applicable.
(1) This university brought me up.
The verb-adverb combination "bring up" has no fewer than thirteen definitions in the full OED. Here are those relevant to the question:
-a.II.27.a To bring into a higher position; to elevate, raise, rear, build up; to raise to a point or amount, etc. See senses of up.
1477 Earl Rivers (Caxton) Dictes 142 Yf he see that fortune raise and bring up som other of lower degre.
1611 Shakes. Wint. T. iv. iv. 544 Your discontenting Father, striue to qualifie And bring him vp to liking.
This definition of bring up is characterized by various synonyms for elevate. It refers us to the different senses of up, indicating that this one definition can have several meanings. The most obvious and current is spatial ("I brought the laundry upstairs"). However, the two quotations of bring up cited above are examples of a state of being. We find two definitions of up which could apply
-b.2.b A rise in life; a spell of prosperity; a success. Usu. pl., and contrasted with down(s). (Cf. up and down n. 2 a.)
-d.2.d A state of mental stimulation or excitement. Cf. high n. 1 h. U.S. colloq.
So, this definition of bring up can easily mean something like "having increased one's life quality/prosperity/successfulness." Seems like exactly the kind of thing one might say about university, so I think the noun definitely works here.
Of course, this would probably be designated as an obsolete definition if the strange verb-multiadverb situation didn't split up the definition. Personally I think the construction still makes sense once considered.
I also included the sense of bring up meaning "made to have one's mood elevated," since it would also work as a correct answer to the question, though from the context I think it's not what the asker had in mind. It's somewhat rare, but certainly a contemporary usage ("Wow, seeing Mary again really brought me up").
(3) This university brought me up in study, culture and my character.
I don't believe this would work. The familiar definition meaning "to rear a child" has some very tempting similarities:
-b.II.27.b **To rear from childhood; to educate, breed. **
1511–2 Act 3 Hen. VIII, iii. §1 To enduce and lern theym and bryng them uppe in shotyng.
1588 A. King Canisius' Catech. 50 Fosterit, teachit, and brocht vp in continuall exercise.
The sub-definition educate is what's interesting here. The quotations are basically the same as the form of your example; however, we must keep in mind that the OED is listing a single definition, and any words listed under it (like educate) must be taken as part of that definition.
So although in the same form, and meaning much the same thing ("to advance one's skill"), these quotations are in the context of the usual sense of bringing up a child. So, this definition of bring up might be rephrased "to rear from childhood immersed in" or "to cultivate a child's skill in," and so cannot be applied to other situations separate from child-raising.
If looking at a different noun than university, perhaps something like boarding school, orphanage or neighborhood friends (something other than parents which could be said to contribute significantly to the rearing of a child), then a very similar example might be technically acceptable! An interesting point to consider.
(2) This university raised those students.
Like bring, raise is another word with a very long entry. OED even sees fit to include a paragraph about it in the headword section, saying "From an early period the word has been extensively used in a great variety of senses, the exact development of which is not always perfectly clear."
a.I.6.a To rouse up, to give or add vigour to (the mind, spirit, etc.); to animate, stimulate.
1728 Pope Dunc. ii. 223 To move, to raise, to ravish ev'ry heart, With Shakespear's nature or with Jonson's art.
b.I.6.b To encourage, inspire (a person) with courage, confidence, hope, etc. Obs.
1652 Needham tr. Selden's Mare Cl. Ep. Ded. 12, I am raised with more than ordinary confidence, that the same Spirit of Justice will carrie you on.
This definition is fairly clear in its applicability; while the definition of bring up was concerned with a thing contributing to important life goals or quality, this would say something much different: the impact of university upon one's mind, energy, vital force.
Although, interestingly enough, we also find a definition narrowly concerned with social/political advancement:
18.III.18 fig. a.III.18.a To promote or advance (a person, people, etc.) to a higher rank, office or position; to exalt in dignity or power.
1667 Milton P.L. xii. 162 A Son whose worthy deeds Raise him to be the second in that Realme of Pharao.
1874 Green Short Hist. ii. §6. 90 Charter after charter..raised the townsmen of boroughs from mere traders..into customary tenants.
This definition could apply to university in several ways; it might improve the social standing or "station" of individual students, or generally improve the sociopolitical condition of those who attend in general, as in the second quotation.