Bernoulli umbra is defined in classical umbral calculus as in Taylor - Difference equations via the classical umbral calculus.

Yu - Bernoulli Operator and Riemann's Zeta Function shows that $\operatorname{eval}\log (B+1)=-\gamma$ (the later paper uses the symbol $B$ for what is defined as $B+1$ in the first paper).

Since we can define Bernoulli umbra as a formal Laurent series in Levi-Civita field as

$$B=\varepsilon^{-1} -\frac{1}{2}-\frac{\varepsilon }{24}+\frac{3 \varepsilon ^3}{640}-\frac{1525 \varepsilon ^5}{580608}+\dotsb$$

and

$$B+1=\varepsilon^{-1} +\frac{1}{2}-\frac{\varepsilon }{24}+\frac{3 \varepsilon ^3}{640}-\frac{1525 \varepsilon ^5}{580608}+\dotsb$$

(the numerators of the terms are given in https://oeis.org/A118050 and the denominators are in https://oeis.org/A118051), the question becomes simple: what's the intuition behind the fact that $\operatorname{st}\log (B+1)=-\gamma$ (in some closure of Levi-Civita field under logarithm operation)?

Here $\operatorname{st}$ means the $0$-th coefficient of the power series and is equivalent to $\operatorname{eval}$ from the first linked paper.