Disclaimer: This is a partial answer given that my background is modeling of the ocean. I hope that some mantle convection modelers can complement this answer.
The question is good but the answer is complex. The short answer is:
No, they are not the same. Simply because computationally it would not make sense.
I will try my best to break it apart and make it as concise as possible.
Preface
As many people have pointed out scales are key. The environmental fluid dynamics problems we try to solve range enormous scales. However, every single motion is described by the Navier-Stokes (NS) equation, from the simplest flow you can think of all the way to the most complex - that includes turbulence (the continuum hypothesis says that the NS equations are valid when the Knudsen number $K_n \ll 1$).
Take a look at the chart below for oceanic processes alone. Temporal scales span 10 orders of magnitude, while spatial scales span 12 orders of magnitude. Presumably mantle solvers would extend the upper bounds on each of these scales.
The question you ask is specifically regarding oceanic grand circulation (OGC) models and mantle convection (MC) models. So according to the chart below, out of all oceanic models, the OGC and the MC models are the closest as far as temporal scales and spatial scales go.
The complexity of the Navier-Stokes equations and the difficulty in solving them
The Navier-Stokes system can be classified as a hybrid elliptic-hyperbolic type for steady flows and a hybrid parabolic-hyperbolic type for unsteady flows (the hyperbolic character comes from the continuity equation).
Where the nature of the equations say the following about each one and their respective numerical difficulties:
Hyperbolic nature is associated with wave phenomena and advective transport:
Fast waves lead to numerical stability restrictions
Nonlinear part of NS is hyperbolic ($\mathbf{u} \cdot \nabla \mathbf{u}$), which is the part of the equation that leads to turbulence.
Possibly one of the most difficult aspect of CFD is the propagation of sharp density fronts which are hyperbolic.
Parabolic nature is associated with diffusion and mass transport:
Boundary layers are governed by parabolic phenomena and are very thin compared to the environment that drives it. Notice the large scale disparity and associated numerical difficulty.
Turbulence, can be modeled from a parabolic perspective and this typically results in stability concerns on the numerical method employed.
Elliptic nature implies instantaneous propagation of information:
For environmental fluid dynamics, the nonhydrostatic pressure is of elliptic nature.
Although, theoretically, any disturbance propagates at infinite speed throughout the domain, numerical iteration sets a finite speed at which information can propagate.
Non-hydrostatic solvers have to invert a Poisson equation which is very computationally
expen sive. In general, for the nonhydrostatic pressure, the 2-d problem requires the solution of a pentadiagonal, while the 3-d problem requires the solution of a
septadiagonal (7 diagonals) (not all near the main diagonal!).
Scales and numerical solvers
So as one can see by now, solving the NS equations numerically is not a trivial matter. Numerical solvers must face concerns regarding accuracy, stability and consistency, which pose constraints on the timestep and grid resolution that one can employ. See this answer regarding different approaches to numerical solvers. Scales are important for numerical solvers because of the nature of the system of the NS equations (described above) and the analytical mathematical techniques available to us to transcribe those equations to computational mathematical language. As it stands, it is impossible to resolve all temporal and spatial scales, so modelers resort to specific techniques (solvers) that apply to the problem (the scales) that they are interested in.
Conclusion
From their website:
The MITgcm (MIT General Circulation Model) is a numerical model
designed for study of the atmosphere, ocean, and climate. Its
non-hydrostatic formulation enables it to simulate fluid phenomena
over a wide range of scales; its adjoint capability enables it to be
applied to parameter and state estimation problems. By employing fluid
isomorphisms, one hydrodynamical kernel can be used to simulate flow
in both the atmosphere and ocean.
and
CitcomS is a finite element code designed to solve compressible
thermochemical convection problems relevant to Earth's mantle.
My guess is they both use different numerical techniques to solve different versions of the Navier-Stokes equations that make the most sense given the scales of the problem that each one aims to resolve.
