Theoretical TCP port limit is 65535 minus your already in use ports. Can a single network card with 10baseT handle 10TB per day across that many ports? Probably not.
Does it affect performance, yes, extra network load requires extra resources like bonded network connections, load balancers, etc. but your software handling those connections will more likely be the cause of performance issues before your network is.
Your chasing a capacity question that only you can solve. It’s a professional engineering problem that’s always going to difficult to solve, but will always be best solved by yourself and a calculator and maybe some performance testing.
Start with a software designed to handle high amounts of network traffic over TCP and you’ll be off on the right foot. As @adonio suggested, rsyslog and syslog-ng are both good at handling large amounts of TCP traffic.
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