Can my institute use my figure and put their name on it for press releases, etc.? I was wondering if my institute can use a figure which I made for a paper that has been published and crop it slightly and use it for press releases, etc., and instead of putting my name underneath the figure, they just put the institute's name.

Is it generally OK for them to do so, or should they give me credit?
It depends. If (1) you once signed a contract that allows the university to do so, and (2) they contacted your publisher to obtain authorization for duplicating the figure, this may be legally allowed. That depends a bit on the laws in your country, which often differ in such matters.

It is certainly bad style, though. Also, PR offices that are careful enough the think of condition (2) will typically also contact you prior to publication of the press release - especially since cropping of the figure occurred. Also: Don't attribute malice to a deed when other possibilities exist. For example, they could have asked you advisor, who said that this would be fine (whether this is OK by the advisor to do so is a different question). Also, replacing your name by the university name could have been the work of an unexperienced intern in the office, etc.

Perhaps you can contact them and ask if this is a mistake. If you want to go the easy route, you could just mention that you are concerned that the figure violates the publisher's copyright and that cropping the image may not be covered by the agreement they gave. In the same mail/call, you could offer help with this potential problem by provide them with the needed source information for a proper "image citation" and helping with obtaining authorization by the publisher (if needed). You could also express genuine concerns that potential future collaborators on research questions related to the figure could use the information of you being the source to establish contact, which would potentially be beneficial for both your career and the future reputation of the university (by strengthening the research done at your institution).