It may seem odd, but there is no legal requirement for an EIC, the part P approved document makes it clear if the work is inspected by the LABC you will only get a completion certificate and not an EIC.
BS 7871 is not law, but it can be used in a court of law, so although BS 7671 says an EIC should be raised, he is not breaking a law by not raising one.
It is also very odd as to who is responsible, if anywhere on his web site, van, paperwork etc, the electrician shows a scheme providers logo, then it is up to him to register any work, other wise it is down to the home owner.
I have questioned the use of some EICR forms as they have the scheme providers logo, even if it states these forms are for people to use when doing an EICR is outside of their scheme membership system. It has their logo, so the electrician needs to notify, can't really understand why any electrician would buy a pad of scheme providers blanks unless he is trying to con the customer as they are a free download from the IET website.
As
@Tony Goddard says the information on both forms is nearly the same, an EIC can only be completed by the person doing the work, so where it is inspected by a third party it is common to use the EICR form, but the inspector is instructed by the LABC not the home owner, and he is told to include any items not complying with the current BS 7671, this was code 4, which for a normal EICR has been dropped, so although using the same form, it is being filled in as if it is a EIC.
The EICR is normally only concerned with danger (C1) and potential danger (C2) or where access is restricted (FI) so for example missing smoke alarms will not fail an EICR as only concerned with the electrics, and if not there they are not an electrical danger. Or fitting SPD as it may cause a danger to installed equipment not having one, but not people, so outside the remit.
Readings are taken so the next EICR done can see if there is any degrade from original, but the inspector can and often does only inspect a percentage of the installation not everything, unless faults are found, OK with domestic one would aim to check all, but the same forms and system is used for commercial premises where nothing is every done by some one not qualified, so the inspector is looking for things which have degraded not some thing done wrong to start with.
If an electrician is a scheme member then he has to follow the membership rules, which would mean he must issue an EIC, and follow nearly everything else in BS 7671, so unless the scheme provider gives him dispensation he could not wire a home to German standards, but a non member could, as nothing forces him to follow BS 7671. (German sockets do not comply with BS 7671).
So as
@Comlec says, much is dependent on if he is a scheme member, personally I think it should be down to the person doing the work to register it where required, but the law says that's only the case with scheme members, if not a scheme member then down to home owner.
As to selling the house, you will be asked if any work requiring registering has been done since 2004 and to produce documents if it has, but lack of documents can be got around by taking out insurance. Odd the IET have said for years that for domestic an EICR should be done every 10 years, but solicitors never seem to ask for them.
And when my mother died I could not find the paperwork, so applied to LABC for duplicates, but was told it would take 4 months and would cost however long it took them to find them, in other words go away we don't what to be bothered with registered work.