What I've found is that whereas "institute" and "institution" differ in Eng. the Gr. ινστιτούτο is used for both. So, yes both should be acceptable and it seems "institution" is the most usual meaning. So, I added it to the accepted answers. Many thanks. There is also another Gr. word: ίδρυμα. :-)
Why isn't the adjective ending o to agree with the noun
Agreement doesn't always mean that the ending is the same.
Most adjectives have three endings: separate ones for masculine, feminine, and neuter, usually -ος -η -ο, but can also be e.g. -ων -ουσα -ον
But some adjectives have two endings; this can be e.g. -ος -ος -ο or -ης -ης -ες.
διμοφιλής belongs to the -ης -ης -ες class and has -ες in the neuter.