Here is the best example from an enlarged search along with some comments. Like Joro's, it has $10$ positive points.

For $a=73513440=2^7\ 3^3\ 5\ 7\ 11\ 13\ 17$ There are $256$ positive solutions to $x^2-y^2=a$ This gives us $256$ positive integer points $(u,v)$ on the curve $u^2-a=v^2$ and also $256$ points on the curve $u^2+a=v^2.$

Similarly, for $b=191923200=2^{12}\ 3^2\ 5^2\ 7^2\ 17$ there are $243$ positive integer points $(u,v)$ on the curve $u^2-b=v^2$ and also $243$ points on the curve $u^2+b=v^2.$

This gives as $8$ ways to pick an ordered pair of integer-square-ordinate-rich quadrics $(f(u),g(u))$, one for $a$ and one for $b.$ The number of coincidences of the type sought are $0,0,0,1,1,2,4,10.$

for $f(u)=u^2+73513440$ and $g(u)=u^2-191923200$ we have $f(x)=z^2$ and $g(y)=(z+1)^2$ for the $10$ triples $[x,y,z]=$ $$ [1174, 16335, 8654], [1578, 16369, 8718], [6761, 17640, 10919], [8097, 18194, 11793],$$$$ [8511, 18382, 12081], [8627, 18436, 12163], [17574, 23965, 19554], $$$$[18353, 24542, 20257], [69351, 71240, 69879], [97051, 98410, 97429]$$ **LATER** also $$ [6423,17512,-10713],[10789,19540,-13781],[39581,42802,-40499],[63974,66015,-64546] .$$ The Magma online timed out on the rank of the elliptic curve. Based on the first $10$ points it is at least $7$ and including the other $4$ it is at least $8.$

I left a program running for the weekend to check similar curves for $a=2^4\ 3^2 a' a''$ with $a'$ a divisor of $2^{8}\ 3^6\ 5^5\ 7^3\ 11^2$ and $a''$ the product of $0,1$ or $2$ of $\{{13,17,19,23\}}.$ I mention the details not because I think they are especially inspired, but so that someone can try better choices. This gave $49896$ values $a_1 \lt a_2 \lt \cdots$ and twice that many quadrics.

I looked for pairs with a good number of (positive integer) matches and kept those with at least $7.$ I'm happy to share the results if anyone wants to analyze them. My non-exhaustive search gave $190$ pairs of curves. Of these $2$ had $10$ matches,$3$ had $9$, $16$ had $8$ and the other $169$ had $7.$ Also, the larger $a$ values were not of much use. Maybe higher powers of the small primes or a few more large ones are needed.

Most of the $190$ examples come from pairs $a_i,a_j$ with $|i-j|$ relatively small. Of them,
$35$ are under $100$, $81$ are under $500$, $130$ are under $2500$, $168$ are under $5000.$ I checked as far as $|i-j|=7500.$ Note that this is $|i-j|$ and not the more natural $|a_i-a_j|.$

Each of the $ 49896$ $a$ values gives two curves. Of the $380$ curves which occur among the examples, $366$ $a$ values and $367$ distinct curves are used. This makes it somewhat remarkable, but still probably a coincidence, that the pair of curves $f(u)=u^2+367567200$ and $g(u)=u^2+369452160$ match up well in two ways: There are $7$ triples $[x,y,z]$ with $f(x)=z^2 \text{ and }g(y)=(z+1)^2$ and $7$ other triples with $f(x)=(z+1)^2 \text{ and }g(y)=z^2.$

**LATER** Actually, that final example has $14$ triples with $[x,y,z]$ with $x,y \gt 0$ and $f(x)=z^2 \text{ and }g(y)=(z+1)^2.$ Of them half have $z \gt 0.$ Magma reports that the rank is $6.$

Both my example and Joro's example with $10$ points $[x,y,z]$ such that $x,y,z \gt 0,$ turn out to also have $4$ more points with $x,y \gt 0 \gt z.$ I've updated the example to show the other $4$ points and rank bound.