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What can be done, following Kato and Fukaya-Kato, is to define a mod $p$ object $\mathcal L(\rho)$ which - if the conjecture on special values are true - encodes whether or not the special values divided by the period specified in the Bloch-Kato conjectures is a $p$-adic unit or not (then $\mathcal L(\rho)$ is a rather weak version of a mod $p$ $p$-adic $L$-function but probably the best you can do in view of the multiple ambiguities in the definition of special values to begin with).

Independently of the truth of the conjecture on special values, it is then true that congruent geometric* Galois representations will have the same $\mathcal L(\rho)$ (so we may denote it $\mathcal L(\bar{\rho})$) provided either 1) we strip $L$-functions of their Euler factors at all primes of bad reduction (that case is mostly formal) or 2) the two Galois representations belong to the same irreducible components of the universal deformation space of their common $\bar{\rho}$ (that case requires some work). The pair $E_3,E_4$ above establish that such a statement is definitely not true without 1) or 2) and a slightly more involved counter-example shows that the requirement that both representations be geometric is also necessary (that is to say, there exist congruent Galois representations belonging to the same irreducible component of the universal deformation space of their common $\bar{\rho}$ but with non-isomorphic $\mathcal L(\rho)$).

In conclusion, yes, congruences between special values coming from congruences between the underlying objects (Kummer congruences, if you want) abound and there is an object $\mathcal L(\bar{\rho})$ a bit like what you wish for, but these are known to exist only under specific hypotheses on the $\rho_i$ and typically require the previous knowledge of strong versions of the conjectures on special values of $L$-functions to be really meaningful.

What can be done, following Kato and Fukaya-Kato, is to define a mod $p$ object $\mathcal L(\rho)$ which - if the conjecture on special values are true - encodes whether or not the special values divided by the period specified in the Bloch-Kato conjectures is a $p$-adic unit or not. More precisely, that element $\mathcal L(\rho)$ belongs to $\mathbb F_p$ and is well-defined modula an element of $\mathbb F^\times_p$ (sic!). The $\mathcal L(\rho)$ is a rather weak version of a mod $p$ $p$-adic $L$-function but probably the best you can do in that level of generality in view of the multiple ambiguities in the definition of special values to begin with.

Independently of the truth of the conjecture on special values, one may wonder if $\mathcal L(\rho_1)$ is equal to $\mathcal L(\rho_2)$ whenever $\rho_1\equiv\rho_2$ modulo $p$. If that is the case, we may denote this common element $\mathcal L(\bar{\rho})$ (remember though that $\mathcal L(\bar{\rho})$ is just an element of $\mathbb F_p$ modulo $\mathbb F^\times_p$). In general, one cannot even do this, as the pair $E_3,E_4$ above establishes.

However, if either 1) we strip $L$-functions of their Euler factors at all primes of bad reduction (that case is mostly formal) or 2) the two Galois representations are geometric* and belong to the same irreducible components of the universal deformation space of their common $\bar{\rho}$ (that case requires some work), then $\mathcal L(\bar{\rho})$ is well-defined. One can find counter-examples showing that the requirement that both representations be geometric is necessary (that is to say, there exist congruent Galois representations belonging to the same irreducible component of the universal deformation space of their common $\bar{\rho}$ but with different $\mathcal L(\rho)$).

In conclusion, yes, congruences between special values coming from congruences between the underlying objects (Kummer congruences, if you want) abound and there is an object $\mathcal L(\bar{\rho})$ a bit like what you wish for, but that object carries very little information, is known to exist only under specific hypotheses on the $\rho_i$ and typically requires the previous knowledge of strong versions of the conjectures on special values of $L$-functions to be really meaningful.

To give a concrete example, there is a mod $3$ $L$-function attached to the common residual representation of the two elliptic curves $E_1:y^2=x^3-7x-6$ and $E_2:y^2=x^3-27x+486$ but in the current understanding of the term, I don't think it is even meaningful to ask whether there is a $p$-adic $L$-function $p$-adically interpolating the special values of the attached modular forms (if I'm not mistaken, the modular form attached to the first one has finite slope at $3$ but not the modular form attached to the second one, so these two forms do not belong to a family for which we know a $p$-adic $L$-function).

Finally, even when all these problems are solved, there is the little problem that congruent Galois representations need not have congruent special values, as was first observed I think by R.Greenberg and V.Vatsal for the pair of elliptic curves congruent modulo 5 $E_3:y^2=x^3+x-10$ and $E_4:y^2=x^3-584x+5444$ (the special value $L(E_3,1)$ divided by the appropriate period is a 5-adic unit while it is not the case of the second). At first sight then, there can be no naive $\bar{L}_{5}(\bar{\rho},s)$ satisfying your desiderata for $\bar{\rho}$ the residual representation $E_3[5]$ (in a way, this phenomenon already appears for characters and lies lurking in your observation that one has to "place some restrictions on what $\rho_1$ and $\rho_2$ can be").

To give a concrete example, there is a mod $3$ $L$-function attached to the common residual representation of the two elliptic curves $$E_1:y^2=x^3-7x-6$$ and $$E_2:y^2=x^3-27x+486$$ but in the current understanding of the term, I don't think it is even meaningful to ask whether there is a $p$-adic $L$-function $p$-adically interpolating the special values of the attached modular forms (if I'm not mistaken, the modular form attached to the first one has finite slope at $3$ but not the modular form attached to the second one, so these two forms do not belong to a family for which we know a $p$-adic $L$-function).

Finally, even when all these problems are solved, there is the little problem that congruent Galois representations need not have congruent special values, as was first observed I think by R.Greenberg and V.Vatsal for the pair of elliptic curves congruent modulo 5 $$E_3:y^2=x^3+x-10$$ and $$E_4:y^2=x^3-584x+5444$$ (the special value $L(E_3,1)$ divided by the appropriate period is a 5-adic unit while it is not the case of the second). At first sight then, there can be no naive $\bar{L}_{5}(\bar{\rho},s)$ satisfying your desiderata for $\bar{\rho}$ the residual representation $E_3[5]$ (in a way, this phenomenon already appears for characters and lies lurking in your observation that one has to "place some restrictions on what $\rho_1$ and $\rho_2$ can be").

Independently of the truth of the conjecture on special values, it is then true that congruent geometric* Galois representations will have the same $\mathcal L(\rho)$ (so we may denote it $\mathcal L(\bar{\rho})$) provided either 1) we strip $L$-functions of their Euler factors at all primes of bad reduction (that case is mostly formal) or 2) the two Galois representations belong to the same irreducible components of the universal deformation space of their common $\bar{\rho}$ (that case requires some work). The pair $E_3,E_4$ above establish that such a statement is definitely not true without 1) or 2) and a slightly more involved counter-example shows that the requirement that both representations be geometric is also necessary (that is to say, there exist congruent Galois representations belonging to the same irreducible component of the universal deformation space of their common $\bar{\rho}$ but with non-isomorphic $\mathcal L(\rho)$)).

Independently of the truth of the conjecture on special values, it is then true that congruent geometric* Galois representations will have the same $\mathcal L(\rho)$ (so we may denote it $\mathcal L(\bar{\rho})$) provided either 1) we strip $L$-functions of their Euler factors at all primes of bad reduction (that case is mostly formal) or 2) the two Galois representations belong to the same irreducible components of the universal deformation space of their common $\bar{\rho}$ (that case requires some work). The pair $E_3,E_4$ above establish that such a statement is definitely not true without 1) or 2) and a slightly more involved counter-example shows that the requirement that both representations be geometric is also necessary (that is to say, there exist congruent Galois representations belonging to the same irreducible component of the universal deformation space of their common $\bar{\rho}$ but with non-isomorphic $\mathcal L(\rho)$).