The mod $2$ cohomology of $\mathrm{SO}_n$ is not an exterior algebra as soon
as $n\geq3$ (there is a unique non-zero element in degree $1$ whose square is
non-zero, think of the case $\mathrm{SO}_{3}$ which is $3$-dimensional real
projective space) while the homology ring is an exterior algebra.
Another example is $K(\mathbb Z,2)$ (aka $\mathbb C\mathbb P^\infty$), its
integral cohomology ring is a polynomial ring on a degree $2$-generator while
its homology ring is the free divided power algebra on a degree $2$-generator.
For the sake of the readers who are not fluent in German, I provide a translation of the German Wikipedia page (link to the revision at the time of posting this answer):
Hermann Künneth (1892-1975) was the son of the high school ("Gymnasium", the highest form of high school) teacher Christian Künneth. Beginning with 1910, he studied mathematics at the Universität Erlangen and the Ludwig-Maximilians-Universität München with a "break" from 1914-1919 where he served in the German army; he was injured twice and was prisoner of war with the British. Künneth was member of the AMV Fridericiana Erlangen, a musically oriented fraternity. His professors in Erlangen were Ernst Sigismund Fischer, Paul Gordan, Max Noether, Richard Baldus and Erhard Schmidt.
1912 he took his first Staatsexamen to become a teacher and 1920 he took his second. In 1920 he became teacher in Bavaria, in particular at high schools ("Gymnasien") in Kronach and Erlangen. He remained in contact with the University in Erlangen, where he got in PhD under the direction of Tietze in 1922 (and was assistant (professor) beginning with 1921). The title of his thesis was "Über die Bettischen Zahlen einer Produktmannigfaltigkeit" - "About the Betti numbers of a product manifold" (where he proved the Künneth formula).
1923 he became assistant (professor) in Berlin; interestingly, he became 1923 also teacher ("Studienrat") in Kronach. As already indicated, the switched to the high school Fridericianum in Erlangen in 1925, where he became Oberstudienrat in 1950 [this would not be a very high position at a high school these days, but I am not sure how it was then].
In 1942 he habilitated in Erlangen and was Privatdozent (a kind of freelancing professor) after that. After he retired in 1957 from his teaching job, he became associate professor in Erlangen. Otto Haupt said about this: "[he] developed an amazing and surprising scientific activity." (at the age of 65)
1964 he got the Bundesverdienstkreuz am Band (the second lowest order of the "Order of Merit of the Federal Republic of Germany"). (See this newspaper article - it says: "His chivalric personality, of clear judgement, emanates human kindness and witty humour.")
Best Answer
In the case of discrete groups, the MV sequence follows from the following argument: if you have $G=H*K$ then you have a short exact sequence of the form: $$0\to \mathbb{Z}G\stackrel{i}{\to} \mathbb{Z}G/H\oplus \mathbb{Z}G/K\stackrel{p}{\to} \mathbb{Z}\to 0,$$ where $i$ sends $g$ to $(gH,gK)$ and $p$ sends $gH$ and $gK$ to 1. The fact that this sequence is exact can be proven combinatorially using the fact that if you take the bases $\{h\}_{h\in h}$ and $\{k\}_{k\in K}$ for $\mathbb{Z}H$ and $\mathbb{Z}K$ then a basis for $\mathbb{Z}G$ is given by the set of all alternating $\textit{words}$ in these bases. This argument can be generalized also the the definition of coproduct of two Hopf algberas given by Agore. If you have Hopf algebras $H_1$ and $H_2$ and $H=H_1*H2$, then choose bases $\{a\}_{a\in A}$ for $H_1$ and $\{b\}_{b\in B}$ for $H_2$ that contain 1. You then get a basis for $H$ by words in these bases: Write $A' = A\backslash \{1\}$ and $B'=B\backslash\{1\}$. We have the following basis for $H$: $$\{1\} \cup \{a\}_{a\in A'}\cup \{b\}_{b\in B'}\cup \{ab\}_{a\in A',b\in B'}\cup\cdots $$
and you can prove in the same way that you get a short exact sequence of $H$-modules given by $$0\to H\to \text{Ind}_{H_1}^{H}k\oplus \text{Ind}_{H_2}^H k\to k.$$ Applying now the relevant functors will give you a long exact sequence, which is the Mayer-Vietoris sequence you need here, using the fact that for a right module $M$ it holds that $\text{Tor}^H_*(\text{Ind}_{H_1}^H k,M)\cong \text{Tor}^{H_1}_*(k,M)$.
I assume here that $k$ is a field for this construction to work.