Does the following voltage regulator have a specific name?
This circuit is trying to be a "linear regulator".  That means it ideally puts out a constant voltage from a higher varying input voltage.  The LM7805 is a common example of a linear regulator all in a 3 pin package.  It can take from about 7.5 to 35 volts in, and make a steady 5 V out.

This particular linear regulator has a pretty high headroom or dropout voltage.  That's the difference between the minimum required input voltage and the output voltage.  It requires the B-E drops of Q2 and Q3, plus whatever headroom the opamp needs between its positive supply and maximum output voltage.  The total could be 2 V or more.  That's quite a lot by today's standards.  Even the old LM7805 only requires about 2.5 V.

This is definitely not a LDO (low dropout) regulator.  I guess you could call it a HDO regulator for fun, but that's not really a name.  Nobody advertises their non-LDO regulators as specifically having high dropout.  They just don't say anything and quietly list the dropout voltage in the specifications.

Q4 with R2 and R3 appears to be a crude attempt at current limiting, but the way it's hooked up won't yield very good results.

You can do better with about the same parts.