this question is related to this one Geometry for Anderson's motives?, though the previous one doesn't answer exactly my question.

Let $\mathbb{C}_{\infty}$ be the function field analog of $\mathbb{C}$ for $\mathbb{F}_q (\theta) = \mathbb{Q}_{\infty}$ and $A = \mathbb{C}_{\infty} [T, \tau]$ the Anderson ring ( i.e., $T$ is central and $\tau$ acts as the Frobenius endomorphism).

An Anderson $T$-motive is simply a left $A$-module $M$ which is free and finitely generated over $\mathbb{C}_{\infty} [\tau]$, and satisfies $$(T - \theta)^n M/\tau M = \{ 0 \}$$ for some $n > 0$. If, furthermore, $M$ is finitely generated over $\mathbb{C}_{\infty}[T]$ (or equivalently, free of finite rank), then it's called an abelian $T$-motive or $t$-motive.

First of all, why the word "motives" in "Anderson $T$-motives"?

Are $T$-motives analogous to motives of arithmetic schemes? What's its number field analogue?

Thanks in advance.