Every language is chock full of synonyms—hundreds or thousands of them.
Usually, those synonyms have some difference. They might:
- … have slightly different connotations (usually "baby" includes toddlers, but "infant" doesn't),
- … be usable in different sets of constructions (compare "eat" and "consume"—only the latter requires an object),
- … be in different registers (compare "hello" and "hi"),
- … differ only in euphony (one sounds better next to certain words but worse next to others),
- … differ only when speaking across dialects (to an English speaker whose dialect has both "eggplant" and "aubergine", they're identical—but having both lets you talk to someone from Los Angeles who only knows "eggplant" and to someone from London who only knows "aubergine"),
- … etc.
But they're still clearly "words for the same notion".
So, to answer your questions:
- How long such a situation can last? Well, not quite forever, because no word lasts forever with the same meaning. Eventually one of the two synonyms will probably shift in meaning until they're no longer synonyms, or die out, or the language itself might die out… but certainly for centuries.
- Is it good for a language to have it? Probably. Otherwise, language evolution would presumably work harder than it does to eliminate synonyms. It's possible that this is just a defect in the human "language organ", or in human society, but it doesn't seem likely. (Also, consider that having a pair of synonyms gives the language the raw material to adapt one of them in a new need arises, without losing the other one.)
- Should language bearers and linguists do something about it? Definitely not. It's not up to linguists to try to forcibly change languages. As for normal speakers, they can try, but (a) there's no good reason to, and (b) it almost never works.1
1. If you're wondering why I said "almost", there is at least one counter-example: Ataturk's modernization of Turkish. But there were a lot of special circumstances there, like the fact that the official language they were modernizing wasn't actually spoken by anyone but the elites.