100 + 100 ≠ 200
First off -- you do not need a 200A service to feed two 100A panelboards, provided the total load as determined by NEC Article 220's calculations does not exceed the service ampacity. This is a consequence of the how 230.90(A) Exception 3 interacts with 230.40 Exception 2 and your specific setup (bold running text mine for emphasis, italics theirs):
230.90 Where Required.
Each ungrounded service conductor shall have overload protection.
(A) Ungrounded Conductor. Such protection shall be
provided by an overcurrent device in series with each ungrounded service conductor that has a rating or setting not
higher than the allowable ampacity of the conductor. A set
of fuses shall be considered all the fuses required to protect
all the ungrounded conductors of a circuit. Single-pole circuit breakers, grouped in accordance with 230.71(B), shall
be considered as one protective device.
Exception No. 1: For motor-starting currents, ratings that
comply with 430.52, 430.62, and 430.63 shall be permitted.
Exception No.2: Fuses and circuit breakers with a rating
or setting that complies with 240.4(B) or (C) and 240.6
shall be permitted.
Exception No.3: Two to six circuit breakers or sets of
fuses shall be permitted as the overcurrent device to provide the overload protection. The sum of the ratings of the
circuit breakers or fuses shall be permitted to exceed the
ampacity of the service conductors, provided the calculated
load does not exceed the ampacity of the service
Exception No.4: Overload protection for fire pump supply
conductors shall comply with 695.4(B)(2)(a).
Exception No.5: Overload protection for 120/240-volt,
3-wire, single-phase dwelling services shall be permitted in
accordance with the requirements of 310.15(B)(7).
(Note that exception 5 has to do with the 310.15(B)(7) allowances for residential service conductor sizing -- they need to be taken into account when determining the total ampacity of your service and the ampacity of your individual sets of service entrance conductors, but stop there.)
A tale of a misplaced panelboard
But, there's more! While what the electrician did (putting a second main panel on your service) would have been OK when done properly as it would fall under 2014 NEC 230.40, exception 2 (bold running text mine for emphasis, italics theirs):
230.40 Number of Service-Entrance Conductor Sets.
Each service drop, set of overhead service conductors, set
of underground service conductors, or service lateral shall
supply only one set of service-entrance conductors.
Exception No.1: A building with more than one occupancy shall be permitted to have one set of service-
entrance conductors for each service, as defined in 230.2,
run to each occupancy or group of occupancies. If the
number of service disconnect locations for any given classification of service does not exceed six, the requirements of
230.2(E) shall apply at each location. If the number of
service disconnect locations exceeds six for any given supply classification, all service disconnect locations for all
supply characteristics, together with any branch circuit or
feeder supply sources, if applicable, shall be clearly described using suitable graphics or text, or both, on one or
more plaques located in an approved, readily accessible
location(s) on the building or structure served and as near
as practicable to the point(s) of attachment or entry(ies) for
each service drop or service lateral, and for each set of
overhead or underground service conductors.
Exception No.2: Where two to six service disconnecting
means in separate enclosures are grouped at one location
and supply separate loads from one service drop, set of
overhead service conductors, set of underground service
conductors, or service lateral, one set of service-entrance
conductors shall be permitted to supply each or several
such service equipment enclosures.
Exception No.3: A single-family dwelling unit and its
accessory structures shall be permitted to have one set of
service-entrance conductors run to each from a single service drop, set of overhead service conductors, set of underground service conductors, or service lateral.
Exception No.4: Two-family dwellings, multi-family dwellings, and multiple occupancy buildings shall be permitted
to have one set of service-entrance conductors installed to
supply the circuits covered in 210.25.
Exception No.5: One set of service-entrance conductors
connected to the supply side of the normal service disconnecting means shall be permitted to supply each or several
systems covered by 230.82(5) or 230.82(6).
However, your electrician screwed up when he put the second panel on the outside of your garage, as that ruins the grouping required by 230.40, exception 2 and 230.72(A):
230.72 Grouping of Disconnects.
(A) General. The two to six disconnects as permitted in
230.71 shall be grouped. Each disconnect shall be marked
to indicate the load served.
So, in any case, you'll need to have the electrician move the second panelboard inside to a spot next to where the first one lives, or replace the first panelboard with an exterior unit that can be mounted next to where the new one lives.