I would like to say "people hold a university degree". Can I paraphrase it as "people earned/possess/own a uni degree"? Also is it possible to say "He attained/achieved a higher degree of education than the others?" Thanks!
There are several questions here.
(1) Can one replace hold a university degree with has earned a university degree?
Yes, the two terms are interchangeable, insofar as one holds a degree if and only if one has earned it (setting aside some highly unusual scenarios). The two terms, however, direct the attention of the audience in subtly different directions. If one uses hold, one is focusing on the present, on the fact that the person has the degree now, and is thus, for example, qualified to take some job. If one uses has earned, one focuses on the past, on the process of working towards the degree (it may thus be a particularly apt term to use if one is describing how hard somebody worked to get the degree).
(2) Can one replace hold a university degree with possess a university degree?
Yes, one would be readily understood if one used possess. It is, however, not clear why one would want to do that; hold is likely to sound better in most contexts.
(3) Can one replace hold a university degree with own a university degree?
No, using own would be awkward, for the reason pointed out by Chappo in the comments. Saying that something is owned would normally be understood to imply that it can be bought and sold.
(4) Can one say 'He attained/achieved a higher degree of education than the others'?
Yes, but is should be noted that degree has a different meaning there than in the other questions. One can speak of 'a higher degree of education' even in a context that deals with informal education, in which no degrees (in the sense in which the word is used in (1)-(3)) are awarded.