A node will not accept or relay a transaction paying less than its minimum relay feerate. The floor exists so that flooding the network with transactions costs something; it is a choice each operator makes, set with -minrelaytxfee, and nothing in the consensus rules requires it.
There are two floors. The configured one, minrelaytxfee, stays put. The effective one, mempoolminfee, rises above it when the mempool reaches its size limit and the node starts evicting its cheapest transactions, and falls back as the pool drains. getmempoolinfo reports both, in BTC per thousand virtual bytes.
The minimum rung on the Network tab is mempool.space's minimumFee: its node's effective floor at that moment, not an estimate of what will confirm.