Does the word "again" have a tense?

The use of "Make [noun] [adjective] Again" means that [noun] was once [adjective], and then was not , and the speaker wants it to be [adjective] in the future.

There isn't, in the mere statement, any implication that the speaker wants to go "backwards". If you were talking about a company that was the biggest in the world, and you wanted to make it the biggest in the world again, that wouldn't mean going backwards necessarily. Remember I am making statements only about grammar.

"Keep [noun] [adjective] Again" implies that [noun] has been kept [adjective] before, and that we want to do it again. It's a bit of doubtful grammaticality in the example you cite (when was the time that America was kept great before?), but for slogans 'nice-sounding' overrides 'grammatical'.

While, looking at this strictly, no tense can be assigned to any parts of speech other than verbs, and so the frequency adverb 'again' has no more a tense than it does sidewhiskers, broadened usages often occur and often become established, acceptable. From the internet I've collected several examples where 'is past tense' is used informally to mean 'is history', 'is no more', 'is confined to the past'. This is a typical metaphorical broadening.

• 'Peak Oil is Past Tense': Exxon, Shell, and Aramco Admit the Peak is Real ... The IEA's Chief Economist Fatih Birol declared: Peak Oil is here, or rather, was here in 2006. And it really doesn't matter whose analysis you prefer....

• a fund for clean energy, oil barons are ... past tense [Google search]

• But beauty parlours are past tense.

• You can’t change yesterday but you can enjoy today and you can determine your tomorrow! All of your yesterdays are gone … they are past tense. [Carol Mcleod]

• 'The hair is gon.' ... Yep, that's the first thing coming up to my mind, when I see good old Willie [Nelson] standing up there on stage at the Circus Krone in Munich. Mind me, his long plaits were something like his brand. But they are past tense now. [MusicMirror_Reviews]

So " 'Again' is past tense" ( for "If we use the word 'again', it means we're revisiting the same old ideas / plans / projects that didn't work too well back in the day") is pithy, witty, and better than acceptable on one level. But not in a linguistics class.