In Quran, the followers of the religious laws of our prophet are referred to as "الذین آمنوا". The phrase is similar to "الذین هادوا" which refers to the followers of the religious laws brought by Moses (PBUH) (and other prophets following him), i.e. Jews.
It doesn't refer to the Jews before the time of our prophet. The Jews at the time of the prophet are also referred to as "الذین هادوا" in Quran (e.g. verse 5:41). So the verse does apply to Jews and Christians during the time of the prophet and after his time.
Note that the verse states that "the followers of Mohammad (PBUH) and those who follow Judaism and Christians and Sabeans whomever believes in God and the last day and does good". That is those following our prophet are added to these other religious groups and the verse states very explicitly that if any of them satisfies those three conditions they shall have neither fear nor sorrow. And These conditions do not include believing in our prophet, do not include following the religious laws given to him. If they accepted and followed the laws brought by our prophet they wouldn't be called Jews, etc.
Those conditions also apply to the followers of the Islamic laws, i.e. simply following the Islamic laws is not considered to result in salvation in this verse. A person following Islamic laws should also satisfy the later three conditions, otherwise this verse doesn't apply to that person.
So what the verse implies that following a particular set of religious laws (Islamic Sharia, Judaism halaka, etc) is not necessarily nor sufficient for salvation. What is important is iman in God and the last day and doing good. (However, there is a catch here: iman in God means submitting to his will and if a person knowingly reject his will and insists on disobeying God's orders then that person doesn't have iman. So if a person from those religions knows that Mohammad (PBUH) is sent from God but rejects him then the verse does not apply to that person because that person does not have iman.)
If being Jew or Christian meant that the person doesn't have iman after our prophet's time we wouldn't have verses in Quran were God allows relations between Muslims and them, eating from each others food, marrying each other, etc. (e.g. verse 5:5). In Quran the treatment of the people of the book is clearly and explicitly different from the non-believers. There are several very positive verses about the people of the book.
One of the strongest related verses is verse 5:48 which states that for each of these religions God has given a (different) set of religious laws and if God wanted he could make all of them one, but he didn't. Therefore he doesn't want so. The preceding verse (5:47) orders that the followers of Christianity should judge between themselves based on Injil, this is during during the prophet's time and even the prophet has to judge between them based on Injil and not based on Islamic laws.
The verse 5:48 ends by ordering the followers of all of these religions to speed and outpace each other in doing good.
So the answers that say the verse is about the time before our prophet do not look consistent with Quran.
Now, let's return to your question. You ask:
"why do people need to embrace Islam" if the followers of the previous religions can also obtain salvation without embracing Islam (under the conditions mentioned in the verse)?
A follower of these religions needs to have iman in Mohammad (PBUH) and what is given to him (Quran) if that person knows that this is from God. A person who knows this has to do so because it is God's explicit order to them, e.g. verse 2:41. It is also part of God's covenant with them to accept the prophets that God sends(verse 5:12). If a person doesn't know these, then that person may not need to accept them, however such a person should neither reject them. A person lacking knowledge should neither accept nor reject. Accepting without knowledge and rejecting without knowledge about a matter are both bad.
So if a follower of these religions does not know that Mohammad (PBUH) is sent by God, that person may not need to accept it, but must not rejects either.
This is what I understand from Quran, and God knows best.
