Well - I don't know. Class names can change, if there are some differences (like C13B) for NEW ships custom-built overseas. For Hamilton class cutter types transferred as older assets, Navies usually do not change class types.
Shadhinota class has significant differences from Jiangdao 056 class, with hull displacement being different (1300 tons vs. 1500 tons for Jiangdao class), superstructure being quite different (especially bridge area profile), sensors and armaments are also Western sourced, some of them. So much so, that Chinese themselves have named it C13B. Some degree of bespoke customization is evident.
The same Jiangdao class was changed for Nigeria and Algeria too I believe? Nigeria version (P18N Centenary class) was with a much larger 1800 ton hull with less armament, more like an OPV. For P18N, P denotes Patrol and N is for Nigeria.
This is 056 Jiangdao class for PLAN - look at bridge area
Now look at Bridge area for C13B Shadhinota Class (C is for corvette, B is for Bangladesh, 13 for 1300 ton displacement)
This is Nigerian P18N - note the hull, much more lengthened compared to the stock 056 class. The bow profile and bridge area are also very different.
You are correct, class is designated by first name of the first ship in a series of similar builds. Unless role changes, then it is a different class. For example the large patrol crafts (LPC) built at KSEW, the ASW versions have a different role and different class name compared to the Large Patrol crafts bought from China. There may be no analogues for those LPC's from PLAN, I don't know of any.