There are two aspects of your question that are inaccurate.

First of all, I have no idea where you are getting the idea that, in Roman legend, Lucretia killed herself to avoid becoming a "war bride," since that is not at all how the story is told in Roman sources.

The Roman historian Titus Livius (lived c. 59 BCE – c. 17 CE) tells the most famous version of the story of Lucretia's suicide in his Ab Urbe Condita 1.57–59. Livius describes how a group of men, including the prince Sextus Tarquinius and a man named Tarquinius Collatinus, are drinking and boasting about how modest and industrious their wives are. They end up going back to Rome in secret to spy on their wives and see what they are doing to see whose wife is really the most virtuous and they find Collatinus's wife Lucretia working hard at her wool while all the other men's wives are partying.

This makes Sextus Tarquinius burn with mad lust for Lucretia because of her beauty and proven chastity. A few days later, he secretly goes to Collatinus's house, where he is received as a guest. Then, that night, he sneaks into Lucretia's bedroom while she is asleep. He wakes her up with his hand on her breast, pinning her down, with a knife at her throat, and he orders her to let him penetrate her, threatening that, if she screams or tries to resist, he will murder her and his own slave and arrange their bodies together so that it will look like she was committing adultery with the slave. Her rapes her and then leaves.

Lucretia immediately summons her husband Collatinus and her father Spurius Lucretius. She explains to them what happened, urges them to punish Tarquinius for having committed adultery with her, and then she kills herself with a knife to punish herself for having committed adultery with him. Livius is very explicit about why Lucretia kills herself in 1.58.9–11. He writes:

". . . mentem peccare, non corpus, et unde consilium afuerit, culpam abesse. 'vos,' inquit, 'videritis, quid illi debeatur: ego me etsi peccato absolvo, supplicio non libero; nec ulla deinde inpudica Lucretiae exemplo vivet.' cultrum, quem sub veste abditum habebat, eum in corde defigit prolapsaque in volnus moribunda cecidit."

This means, in my own translation:

"[Collatinus and Lucretius were saying to her] that the mind errs, not the body, and that where there was no intent, guilt is absent. 'Let you,' she said, 'decide what is owed to that man [i.e., Sextus Tarquinius]. For my own part, although I absolve myself for the mistake, I do not free myself from the punishment. Never anytime in the future will any unchaste woman live through the example of Lucretia.' She plunged the knife, which she was holding concealed underneath her clothes, into her heart and, sliding it forward in the wound, dying, she fell."

The moral of the story as Livius tells it is that no woman who has been sexually penetrated by any man other than her lawful husband should be allowed to live under any circumstances, even if she was raped. Accordingly, the ideally chaste married woman, if a man other than her husband rapes her, should kill herself, rather than dishonor her husband by continuing to live. It is a profoundly misogynistic story.

The Greek historian Dionysios of Halikarnassos (lived c. 60 – c. 7 BCE) tells a version of the story of Lucretia that is very similar to Livius's, but lacks Lucretia's explanation of her suicide, in his Roman Antiquities 4.64–70. The later Greek historian Kassio Dion (lived c. 155 – c. 235 CE) tells another version of the story in his Roman History 2.11 that is also very similar to Livius's and that includes a very similar explanation of her suicide that is almost certainly based on Livius's own.

That's one inaccuracy in your question. The other inaccuracy is that, in the ancient world, it was far more common for soldiers from conquering nations to take women from conquered nations as slaves (with, of course, the expectation that the women they took as slaves would serve them sexually), rather than "war brides." This may sound like a pedantic distinction, since you may think a woman who is forced to marry a man who has absolute power over her and whom she must obey is effectively a slave, but, in the status-obsessed ancient Mediterranean world, the distinction between a man's wife and his slave was an important one.

Generally speaking, being a man's wife was a position reserved for a "respectable" free woman, not for a woman the man had taken in war. Agamemnon famously describes the position of Chryseïs, a young woman whom the Achaians have captured in war and whom he has claimed as his slave, in the Iliad 1.29–31:

"τὴν δ᾽ ἐγὼ οὐ λύσω: πρίν μιν καὶ γῆρας ἔπεισιν
ἡμετέρῳ ἐνὶ οἴκῳ ἐν Ἄργεϊ τηλόθι πάτρης
ἱστὸν ἐποιχομένην καὶ ἐμὸν λέχος ἀντιόωσαν:"

This means, in my own translation:

"And I will not release the girl until even old age comes to her
in my house in Argos, far away from her father,
as she paces before the loom and she serves me in my bed."

Later, in the Iliad 19.291–300, Briseïs, another young woman whom the Achaians have taken captive, who was originally given as a slave to Achilleus, describes how she hoped that Achilleus would free her and make her his wife. It is, of course, highly dubious whether any historical women ever hoped this, but it does show clearly how being a man's wife was thought of as a higher status than his slave woman.

With all that preface out of the way, it is certainly the case that it was common for women (and men) in the ancient Mediterranean world to kill themselves in order to avoid capture, humiliation, and possible enslavement.

The most famous case is, of course, Kleopatra VII Philopator, the queen of Egypt, who killed herself in August 30 BCE to avoid being captured by Octavian and humiliated in his triumphal procession. She is reported to have said "οὑ θριαμβεύσομαι," which means "I will not be led in a triumph." The most popular story today holds that she killed herself by allowing an Egyptian asp to bite her on the breast, but, as I discuss in this blog post I wrote back in 2019, it is much more likely that she either drank poison or poisoned herself through a deliberate cut on the arm.

In this particular case, it is highly unlikely the Octavian would have taken Kleopatra as his personal slave, especially since she was a queen and she would have had enough supporters still around that keeping her alive in any capacity would have been foolish. He was most likely planning to execute her after his triumph in the same way that Julius Caesar executed Vercingetorix after his triumph in 46 BCE.

There is also evidence for men killing both their wives and themselves in the wake of a defeat to avoid capture and enslavement. The famous Ludovisi Gaul or Galatian Suicide statue group, which is believed to have originally been created by the sculptor Epigonos of Pergamon at some point between c. 230 and c. 220 BCE to celebrate the victory of Attalos I of Pergamon over the Galatians, but which survives only through a later Roman copy of the early second century CE, depicts a Galatian man who, in the aftermath of the defeat, has just killed his wife and is about to kill himself so that the Pergamenes will not be able to capture them and take them as slaves.

Similarly, the Jewish historian Titus Flavius Josephus (lived c. 37 – c. 100 CE) describes in his book The Jewish War 7.9.389–406 how, in 73 CE, when the Romans took the fortress of Masada, rather than be captured, the men defending the fortress killed their own wives and children and then drew lots to kill each other so that none of them would survive for the Romans to capture them and take them as slaves.

In India, there is a historical practice known as jauhar, in which a woman or group of women kill themselves to avoid being captured, taken as slaves, or raped by conquering soldiers. I do not know much about how common this practice actually was, but I'm sure an expert in premodern Indian history could tell you more.