This is an open question in SLA, which even further depends on a person's fluency in a given L2, although most would agree at least some similarities are shared.
At some level, certain processes are shared between L1s, and L2s, as the answer to the question linked in this question states. Broca's and Wernicke's areas, regions of the brain responsible for speech production and comprehension, respectively, are activated in both L1 and L2 processing, although there are some differences. (Not only that, they are used in signed languages as well.)
This is studied through the acquisition of features of a second language (gender, case marking, definiteness, etc.), and how well learners use them. Think of stereotypical errors like Russians dropping "the" and "a" in English.
There are two main schools of thought.
Some believe that yes, learners can fully acquire features in an L2. This is known as the Full Transfer Full Access hypothesis. Learners' struggles with certain features is accounted for with the Missing Surface Inflection Hypothesis, which states that L2 errors are not due to a representational/syntactic deficit, but rather, mistakes in mapping morphological/lexical components to the syntactic representation.
The other school of thought states that, beyond the critical period (about age 4), L2s are not processed in the same manner as L1s. If you support the UG, this is the belief that the UG is not accessible - or at least not fully accessible - after the critical period. This is accounted for by the Fundamental Difference Hypothesis, which simply states L2 linguistic systems are fundamentally different from L1. The Representational Deficit Hypothesis is how followers of this school of though account for morphological errors, which states that learners cannot acquire features missing from their L1 (for example, English speakers learning Spanish grammatical gender). However, to account for learners that do acquire missing features, the RDH states that L2 learners can use their L1 grammar to approximate L2
Language processing can be studied through a number of neuroimaging methods, primarily functional magnetic resonance imaging (fMRI) and electroencephalography (EEG). EEGs can be used to measure Event Related Potentials (ERPs), which are located by averaging neurological responses following a type of stimulus, which allows researchers to extract the signal from the noise).
ERPs do not tell us what is being activated - the electrical signals are smeared all over - but can be used as evidence that the same process is occurring between stimuli/individuals.
ERP studies have identified two major responses to linguistic anomalies: The N400 (negative-going, 400 ms after stimulus) and the P600 (positive-going, 600 ms), which occur in response to semantic and syntactic errors, respectively.
For example, the N400 (in native speakers) would occur after the word bake in the sentence below (since cats can't bake, but subject-verb agreement is OK).
# The cat bakes the cake.
However, the P600 occurs after structural errors, for example, below:
* The cat eating the cake.
(Osterhout, L., et al. Second-language learning and changes in the brain. Journal of
Neurolinguistics (2008), doi:10.1016/j.jneuroling.2008.01.001)
L2 learners, however, start off with N400 (semantic) responses to anomalies like the second one. This is because they rely on declarative memory to identify words in context, rather than procedural memory to break eating down into
eat + ing.
In other words, they first learn all the forms of a verb by heart (eat, ate, eating, eaten), then, as they gain fluency, begin to use syntactic procedures to parse a sentence.
To summarize, early L2 learners process some elements of their L2 differently, relying on memorizing words to understand. More fluent L2ers begin to acquire native-like processing abilities, although the timing and onset of those abilities vary with demographics such as age, age of acquisition, innate ability, motivation, etc.
Full Transfer, Full Access: Schwartz B and Sprouse R (1996) L2 cognitive states and the Full Transfer/Full Access model.
Second Language Research 12: 40–72.
Missing Surface Inflection: Haznedar B and Schwartz B (1997) Are there optional infinitives in child L2 acquisition? In: Hughes
E and Greenhill A (eds) Proceedings of the 21st annual Boston University Conference on
Language Development (BUCLD) September 1996. Boston, MA: Somerville: Cascadilla Press,
Fundamental Difference: Bley-Vroman R (2009) The evolving context of the Fundamental Difference Hypothesis. Studies
in Second Language Acquisition 31: 175–98.