Google can offer this because you are using a single, fairly unified, ecosystem and they own both your account and browser. However, this ecosystem does not offer interoperability so you are effectively locked-in.
Ironically, this feature is exactly why I can't use Chrome or login on accounts on Android applications: this will auto-sync everything and log you into Google, make logging out very hard, enroll your devices on Google Workspace if that's a professional account, ...
Vendor lockin is where the vendor does things to lock you in, such as "Google made it so my Chromebook can only run Chrome", or "Google made it so my GMail only works in Chrome". Lockin generally is when people other than the vendor do things to lock you in, "Meta made WhatsApp web only work in Chrome", "My bank website only works in Chrome".
"The vendor made a more integrated, better, product and I prefer it" is not lockin of any kind, that's ordinary competition.