The short answer is Yes.

Here are some of the details. You start with a crystal for the representation.
You don't need to specify which model you are taking. When you take a tensor
product a subset of the vertices of the crystal give highest weight vectors.
The rule is a simple rule depending on the depth (or rise) of the vertex.

For the fundamental representation this is particularly straightforward.

If you want to get started with crystals, a good place to begin is:

MR1881971 (2002m:17012)
Hong, Jin ;  Kang, Seok-Jin .
Introduction to quantum groups and crystal bases.
Graduate Studies in Mathematics, 42. American Mathematical Society, Providence, RI,  2002. xviii+307 pp. ISBN: 0-8218-2874-6 

This does not discuss the tensor product rule you asked for. The original reference
for for the general tensor product rule is:

MR0225936 (37 #1526)
Parthasarathy, K. R. ;  Ranga Rao, R. ;  Varadarajan, V. S.
Representations of complex semi-simple Lie groups and Lie algebras.
 Ann. of Math. (2)  85  1967 383--429.

This tensor product rule requires further notation. The advantage is that multiplicities
are calculated directly without any cancelation.