Relationship between LCGs and LFSRs

They're both linear which is of course a weakness from the cryptographic point of view. LCG is $$x_{t}\equiv (a x_{t-1}+c) \pmod n \qquad (1) $$ while LFSR is $$x_{t} \equiv (a_1 x_{t-1}+ a_2 x_{t-2}+\cdots+ a_L x_{t-L}) \pmod 2\qquad (2) $$

One could devise an $L$ term LCG $$x_{t}\equiv (a_1 x_{t-1}+a_2 x_{t-2}+\cdots+ a_L x_{t-L}) \pmod n$$ which could then be reduced mod 2 (both the $x_t$s and the $a_i$s reduced mod 2) to correspond to an LFSR, but that's a bit artificial, since one must pick a large size for $n$ and choose its properties (e.g., is $n$ prime, semiprime, is $\gcd(a,n)=1$) to get a relatively strong recurrence. So this would be overkill, with no appreciable gain in strength compared to (1) and a more complicated choice of the constants $a_i$ would be needed.

And the basic attacks on them are based on quite different ideas, for example the truncation attack on LCGs by Lagarias et al has nothing to do with Berlekamp Massey attack.

I am not aware of any other important similarities between them; at least I can't think of any off the top of my head.