WhatsApp uses your phone as the primary message store. Its architecture is more similar to SMS than to typical DB-driven client-server apps. Their servers are only designed to pass messages through, not persist them (they persist them only until they're delivered to all recipients, which usually is very short). This is an uncommon design these days, but it's also what's let them serve half the world with 5 backend engineers (!) back before Facebook/Meta bought them. It's also why, if you didn't make any backups, you lose your WhatsApp message history when you switch phones.
So the web/desktop clients are thin clients over that store on your phone, which keep an eventually-consistent cache of the messages somewhere locally (indexeddb or something like that I bet). So if I'm not mistaken, when you open the web client, it connects to a WhatsApp relay server which wakes up the WhatsApp app on your phone, asks it for any new messages since last time it connected, and syncs them over. This is a fast but not instant process and what you're seeing is the UI realtime updating as the messages are being synced. The longer since you've had the web client open, the longer this takes.
I'm not sure if it's deliberate but I can totally imagine it to be. It very clearly shows the syncing taking place. Personally, since I'm rather enamored by their design, I love seeing it play out in realtime.
Note, I'm not 100% sure about any of the above, notably core aspects of the architecture might've changed since Meta took over.
PC is independent from phone now, but from observation, it seems like it picks a "main" device and syncs the other ones to that (Inconsistently, probably whichever you used latest)
I can see how ocasionally I'll get a message way sooner on PC, respond, then pick my phone and that will "animate" messages in my phone, so it works both ways
So the web/desktop clients are thin clients over that store on your phone, which keep an eventually-consistent cache of the messages somewhere locally (indexeddb or something like that I bet). So if I'm not mistaken, when you open the web client, it connects to a WhatsApp relay server which wakes up the WhatsApp app on your phone, asks it for any new messages since last time it connected, and syncs them over. This is a fast but not instant process and what you're seeing is the UI realtime updating as the messages are being synced. The longer since you've had the web client open, the longer this takes.
I'm not sure if it's deliberate but I can totally imagine it to be. It very clearly shows the syncing taking place. Personally, since I'm rather enamored by their design, I love seeing it play out in realtime.
Note, I'm not 100% sure about any of the above, notably core aspects of the architecture might've changed since Meta took over.