To my knowledge, one can only make this analogy fully consistent with Weibel's homotopy invariant $K$-theory $KH$ and $G$-theory (although the proofs of what I claim below rely heavily on our understanding of classical algebraic $K$-theory). Then, using the canonical map $K(X)\to KH(X)$, the pairing relating $KH$ and $G$ induce the pairing relating $K$ and $G$ which fits in the folkloric description of Poincaré duality relating $K$ and $G$.

Indeed, classical Poincaré duality is a particular instance of Grothendieck-Verdier duality (i.e. Grothendieck duality for ordinary sheaves of abelian groups). Indeed, if $D(X)$ denotes the derived category of sheaves on a (nice locally compact) space $X$, then we have, for any continuous map $f:X\to Y$ a pullback functor $f^*:D(Y)\to D(X)$ which has a right adjoint $f_*: D(X)\to D(Y)$, and there is push-forward functor $f_!:D(X)\to D(Y)$, the right derived functor of the direct image with compact support functor, which has a right adjoint $f^!: D(Y)\to D(X)$. There is natural map $f_!\to f_*$ which is invertible, and so on.

Now, if $X$ is a space and $a:X\to \{pt\}$ denotes the canonical maps to the point,

 - the cohomology of $X$ with coefficients in $\mathbf Z$ is $a_* a^*(\mathbf Z)$
 - the cohomology with compact support is $a_! a^*(\mathbf Z)$
 - Borel-Moore homology is $a_* a^!(\mathbf Z)$
 - Homology is $a_! a^!(\mathbf Z)$

For nice enough spaces (e.g. algebraic varieties), these are perfect complexes of abelian groups (whence dualizable objects in the derived category of abelian groups), and taking the dual in the derived category exchanges $*$ and $!$. In particular, the dual of homology $a_! a^!(\mathbf Z)$ is cohomology $a_* a^*(\mathbf Z)$, while the dual of homology with compact support $a_! a^*(\mathbf Z)$ is Borel-Moore homology $a_* a^!(\mathbf Z)$. Poincaré duality consists in identifying, when $X$ is smooth complex orientable of dimension $d$, $a_! a^* (\mathbf Z)$ and $a_! a^!(\mathbf Z)(-d)[-2d]$ (where $A(-n)=A\otimes H^2(\mathbf P^1(\mathbf C),\mathbf Z)^{\otimes n}$).

Using Morel-Voevodsky's motivic stable homotopy category $SH$, we can extend this to schemes: to simplify, I will restrict to schemes of finite type over a field $k$. Then, given a a commutative motivic ring spectrum $E$ in $SH(k)$, we may define $D(X)$ as the category of $E$-modules in $SH(X)$ (to be precise, of $a^*(E)$-modules in the $(\infty,1)$-category $SH(X)$, where $a:X\to\mathrm{Spec}\, k$ denotes the structural map). And we have most of the features above, replacing $\mathbf Z$ by $E$ (e.g. we have cohomology $a_* a^*(E)$ and so forth). In the case where $E=KGL$ is the object which represents Weibel's $KH$ in $SH$, this gives a context in which Grothendieck's six operations apply. 

Here, Poincaré duality identifies $a_! a^* (KGL)$ and $a_! a^!(KGL)(-d)[-2d]$ (for $X$ smooth of dimension $d$). Dually, it corresponds to

$$a_* a^!(KGL)\simeq a_* a^*(KGL)(d)[2d]$$ 

but Bott-periodicity also says that $KGL\simeq KGL(d)[2d]$.

Furthermore, for possibly singular $X$, one can check that the global sections of $a_* a^!(KGL)$ really give back $G$-theory:

$$\Gamma(\mathrm{Spec} \,k,a_* a^!(KGL))=G(X)$$
(this is essentially a reformulation of $K$-theoretic Poincaré duality as formulated in the question above, of Quillen's localization theorem for $G$-theory and of homotopy invariance for $G$-theory). And we have:

$$\Gamma(\mathrm{Spec} \,k,a_* a^*(KGL))=KH(X)$$

We could define homotopy invariant $K$-theory with compact support:
$$KH_c(X):=\Gamma(\mathrm{Spec} \,k,a_! a^*(KGL))$$
and $KH$-homology as $\Gamma(\mathrm{Spec} \,k,a_! a^!(KGL))$.

As for the coniveau spectral sequence (a.k.a the motivic Atiyah-Hirzebruch spectral sequence), applied to $G$-theory, Marc Levine has showed that the $E_2$-term will be motivic cohomology defined through Bloch's cycle complexes (also for $X$ singular): this is why the cohomology defined through Bloch cycle complex should not be called "motivic cohomology" but rather "motivic Borel-Moore homology".

Rationally, $KGL$ is naturally a $H\mathbf Q$-algebra, where $H\mathbf Q$ denotes the $\mathbf Q$-linear motivic Eilenberg-MacLane spectrum. In fact, $KGL$ becomes the free Bott-periodic $H\mathbf Q$-algebra. Furthermore, the category of $H\mathbf Q$-modules in $SH(X)$ is then equivalent to $DM(X,\mathbf Q)$ (the category of motivic sheaves over $X$), and the change of scalars functor from $H\mathbf Q$-modules to $KGL$-modules commutes with the six operations (at least if we restric to compact objects). In particular, one recovers Poincaré duality in $KGL$-modules from the one in motives, but with a "Todd-twist": the classical formulations of Poincaré duality as above involve Thom isomorphisms, which themselves rely on a choice of an orientation. Poincaré duality on homotopy $K$-theory as described above corresponds to the orientation of $KGL$ defined by the multiplicative formal group law, while the one coming from seeing rationalized $KGL$ as a $H\mathbf Q$-algebra corresponds to the additive formal group law. Relating the two through an explicit isomorphism is exactly the purpose of Grothendieck-Riemann-Roch theorems.