What is the reaction mechanism between boric acid and 2-ethyl-1-hexanol
The research paper sighted in the question (Ref.1) is about the extraction of boron (as boric acid: $\ce{H3BO3}$) from salt lake brine using 2-ethylhexanol/kerosene as organic phase. It is true that the given diagram in this paper (as shown in the question) is showing an esterification, but the authors refer it to two references, one of them is Ref.2. However, the association used to calculate the distribution of boric acid between aqueous and organic phases in Ref.1 is not the esterification. Accordingly to the paper, the association reaction is as follows:
$$\ce{H3BO3 + nROH &lt;=&gt; H3BO3.nROH}$$
Since there are no water molecule elimination during the reaction, this is not an esterification. It is a solvolysis reaction as OP correctly recognized.
Extraction of boron from existing sources is a challenging problem (Ref.3). Boron can be extracted as neutral ester $(\ce{B(OR)3})$, as boric acid complexes $(\ce{(H3BO3)_x.(ROH)_y})$ or as tetrahydroxy borate $(\ce{R+B(OH)4-})$. However, the extraction of boric acid by monohydric alcohols is caused by purely physical actions and the amount of extracted boric acid is roughly proportional to the concentration of an extractant (Ref.3). Accordingly, it is safe to say the extraction process is based on complexation of boric acid with intended alcohol.

References:

Ran Zhang, Yingming Xie, Jianfeng Song, Lixin Xing, Dingfeng Kong, Xue-Mei Li, Tao He, &quot;Extraction of boron from salt lake brine using 2-ethylhexanol,&quot; Hydrometallurgy 2016, 160,  129–136 (DOI: https://doi.org/10.1016/j.hydromet.2016.01.001)(PDF).
Jianhua LÜ, Jidong LIU, Yujie SUN, Chunli LI, &quot;Kinetics of Forward Extraction of Boric Acid from Salt Lake Brine by 2-Ethyl-1,3-hexanediol in Toluene Using Single Drop Technique,&quot; Chinese Journal of Chemical Engineering 2014, 22(5), 496-502 (DOI: https://doi.org/10.1016/S1004-9541(14)60059-8).
Adam Balinski, Volker Recksiek, Norman Kelly, &quot;Solvent Extraction of Boric Acid: Comparison of Five Different Monohydric Alcohols and Equilibrium Modeling with Numerical Methods,&quot; Processes 2021, 9(2),  398 (15 pages) (DOI: https://doi.org/10.3390/pr9020398).