Signet is more comparable to testnet than it is to regtest.
Regtest is for private use and testing things. Even if signet were available, it is still useful to have your own blockchain for testing where you can generate coins, blocks, forks, and reorgs at will. It allows tests to run quickly and for a specific set of conditions to be tested. There is no need to rely on outside parties, or even having a connection to the internet at all. This makes regtest extremely useful for development. Bitcoin Core's regression tests use regtest (hence the name, regtest is short for regression testing network) and will continue to use regtest.
Signet is a public network where blocks will be produced at some interval by the signer. The signer determines at what rate blocks are produced, what transactions are included in them, whether there are forks, and whether there are reorgs. As a user, you have no control over blocks whatsoever, unlike regtest, where you have absolute control. So this is more comparable to testnet, which is the current public testing network. Both signet and testnet will allow you to use a blockchain where someone else is going through the work of producing blocks. This gives you a more realistic simulation of mainnet.
The main difference between signet and testnet is that signet will have blocks be produced more evenly distributed (like they are on mainnet), and even have planned reorgs to allow you to test your software under these conditions. The main issue with testnet is that due to its difficulty reset behavior, blocks are mined very quickly and often clustered together. So there are usually large gaps between blocks, followed by several (thousand) blocks mined in a short timespan. Signet avoids this issue.
Lastly, you can use different signets by simply choosing someone else to be the signer. There is no need to create a new genesis block and define a new network, you can use the same rules, just a different signer. This allows you to test different scenarios with almost no work at all, just choose a signer that is doing some pattern of blocks that you want to test.