While @Shalom's answer is a start, I don't think it fully explains the topic at hand.
tl;dr: As @Shalom pointed out, there are witnesses involved in the passage, essentially making the "tokens of virginity" (a cloth with or without blood on it) unnecessary. However, there is a dispute in the gemarra which sounds like there is an opinion that it's all about the blood. However, as will be explained, that opinion can be understood to agree that there are witnesses.
What you are asking about is called the parsha/passage of Motzi Shem Ra, someone who spreads a bad name about their wife. As you quoted, it's found in Deuteronomy 22:13-19. The verses describe someone who after completing marriage to his wife (known as nisuin), claims she cheated on him after they had begun the process of marriage (the stage known as erusin or kiddushin). The custom was that between the two stages of marriage, kiddushin and nisuin, there was a period to get ready for marriage (the custom then was 12 months for a virgin and 30 days for a widow, see here). In the interim they were fully married, but didn't live together. If she committed adultery during this time, she would be liable to the death penalty. So essentially this guy is claiming his wife deserves the death penalty.
Verses 20 and 21 which you quote are if he is proven right. The preceding verses deal with if he is proven wrong; how he is punished for slandering her good name. Now, the details of this passage are discussed in Kesubos 44a - 46a. The relevant parts to your question are found on Kesubos 46a. There you'll find a dispute between Rabbi Eliezer ben Yaakov (REbY) and the Sages. REbY says that the passage is only discussing if when they completed their marriage, known as nisuin, they consummated their marriage. The Sages say even if they hadn't consummated their marriage, the passage applies. The Sages explain the passage is a case where the husband presents witnesses who testify that she committed adultery during erusin and is thus liable to the death penalty.
The gemarra points out that the simple reading of the verses fits better with REbY, and the Sages have to justify their position by re-explaining the verses. I won't explain them all, but some important ones are:
Verse 14 :
וְלֹא־מָצָ֥אתִי לָ֖הּ בְּתוּלִֽים
I didn't find in her signs of virginity
The custom used to be to use a cloth during intercourse to prove whether there was blood or not, indicating if she was a virgin or not (see Kesubos 6b and 12a). The simple reading of the verse then is he is claiming he didn't find blood when they consummated their marriage. He claims that she lost her virginity during erusin and thus she would be liable to the death penalty. The gemarra explains that the Sages read the verse as saying "I didn't find witnesses attesting to her virginity", meaning witnesses which could contradict his own, who are claiming she committed adultery.
וְאֵ֖לֶּה בְּתוּלֵ֣י בִתִּ֑י
These are the signs of virginity of my daughter
The simpler reading of the verse is the father presents the cloth which shows the blood of his daughter, proving she was a virgin when they consummated the marriage. The Sages explain the verse means the father produces witnesses who contradict the husband's witnesses.
וּפָֽרְשׂוּ֙ הַשִּׂמְלָ֔ה לִפְנֵ֖י זִקְנֵ֥י הָעִֽיר
They spread the cloth before the elders of the city
Here we see the first mention of the cloth. The simple reading is as above, the father presents the cloth to the elders. The Sages read the verse metaphorically, that the witnesses clarify the matter "like a new cloth". I assume this means it's as clear as a new cloth is clean, but honestly I don't really get the metaphor.
Each of these two explanations of the passage have their advantages and disadvantages. The Sages explanation, while not fitting with the simple reading of the verses, consistently uses witnesses, which is the main form of proof in Jewish law. Cloths aren't proof enough to administer the death penalty. Only witnesses can. Therefore, their explanation avoids this issue.
The question is how to understand REbY with regards to the rest of the Torah. His explanation fits more with the reading of the verses (he even says explicitly, the verses are to be read literally), but why would she get the death penalty if he shows a cloth without blood? Why is she killed if she can't produce a blood-soaked cloth? Tosafos and others are bothered with this question.
Based on this question, Malbim, Torah Temimah, and I'm sure others offer the following explanation of the passage: Even though it's not mentioned in the passage, everyone agrees there are witnesses involved. The Sages explain the whole passage as just an example of a case where he slandered his wife, where he said they consummated the marriage. But the Sages say that even if they never consummated the marriage, and he brought witnesses, the verses can make sense. REbY holds the verses have to be understood literally. Therefore, they really did consummate the marriage. After he did, he didn't find in her signs of virginity. This caused him to investigate the matter and find witnesses attesting to her infidelity. After this there's really no dispute. His bringing the cloth is just to strengthen his claim, but it's really all about the witnesses. When the father brings his own witnesses to discredit the first, the matter is all cleared up like a new cloth1.
To answer your question if this passage is still observed, besides the fact that the death penalty or fines are no longer enforced, the whole case will rarely ever be relevant. Like I wrote at the beginning, the custom was to have a gap between erusin and nisuin. The custom today is that erusin and nisuin are done immediately one after the other (Rema EH 55:1)2. Therefore, it is unlikely that a case would occur that a husband could claim his wife cheated on him during the erusin period. Adultery after nisuin is a different question, unrelated to the passage above.
1 The Malbim wants to use this explanation to answer the question everyone has on the Rambam, who rules like REbY that the passage only refers to when they actually consummated the marriage, but explains the cloth as a simile and not literally, which would sound like the Sages explanation. According to this there's no contradiction.
2 This makes many sugyos in the first chapter of Kesubos no longer relevant. For example, the Gra (ad.
loc. s.k. 11) explains that the first mishnah and daf 9 are no longer relevant.