There is no Play Store app and there is not going to be one. Google does not let real-money gambling brands publish in the EU store without a local licence. What LyraBet ships instead is a PWA, which is the site dressed up to behave like an app once you save it to the home screen.
On Android the install prompt usually appears in Chrome after you visit the site twice. If it doesn't, open the three-dot menu and tap Add to home screen. On iPhone it lives behind the Safari share sheet. Either way it takes a few seconds.
After three weeks on a Galaxy A54 and an iPhone 13, I kept opening the PWA instead of the browser. No repeated login, working push notifications, cold start under two seconds on both phones.
When the install prompt failed to appear, it was usually old Chrome or MIUI blocking the prompt on a Xiaomi phone. Updating Chrome fixed my test handset.