Group by 2 variables, with unique colors for one and unique shapes for the other

You may have to ditch gscatter and do this yourself. From the documentation item on multiple grouping variables:

Alternatively, g can be a cell array containing several grouping variables (such as {g1 g2 g3}), in which case observations are in the same group if they have common values of all grouping variables.

i.e. it can't handle independent variables with independent colours and shapes.

You can use fingroups and some indexing to build up groups and looping over the years for plotting. Here is a robust solution which handles cases like having more groups than colours/markers:

% Define colours and markers
colors = lines(8); colors(8,1)=0.5;
markers = {'x','d','o','+','*','s','p''h'};

% Create colour matrix
[gInputs, uInputs] = findgroups(Inputs);
if max(gInputs) > size(colors,1)
    warning( 'More inputs than possible colors, colors will be re-used' );
end
colors = colors(mod(gInputs-1,size(colors,1))+1,:); % mod to handle out of range case
% Create marker array
[gYears, uYears] = findgroups(Years);
if max(gYears) > numel(markers)
    warning( 'More years than possible markers, markers will be re-used' );
end
markers = markers(mod(gYears-1,numel(markers))+1); % mod to handle out of range case

figure(); hold on
for iYr = 1:max(gYears)
    idx = (iYr == gYears); 
    scatter(t1Response(idx), t2Response(idx), 20, colors(idx,:), 'Marker', markers{iYr}, 'displayname', num2str(uYears(iYr)), 'LineWidth', 2 );
end
hold off
legend('show')

If you want the legend to reflect the combinations of year and input then you will have to use a double loop

% Setup as above...
figure(); hold on
for iYr = 1:max(gYears)
    for iIn = 1:max(gInputs)
        idx = (iYr == gYears) & (iIn == gInputs); 
        scatter(t1Response(idx), t2Response(idx), 20, colors(idx,:), 'Marker', markers{iYr}, 'displayname', sprintf('%d: %d',uYears(iYr),uInputs(iIn)), 'LineWidth', 2 );
    end
end
hold off
legend('show')

You could probably make the last example quicker using line instead of scatter but the syntax is slightly different so I've left it to avoid complication.

To get a more concise legend, you would have to spoof some lines with the formats you want. I've restructured the code a bit to show how this can be done:

% Define colours and markers
colors = lines(8); colors(8,1)=0.5;
markers = {'x','d','o','+','*','s','p''h'};

% Create colour matrix
[gInputs, uInputs] = findgroups(Inputs);
if max(gInputs) > size(colors,1)
    warning( 'More inputs than possible colors, colors will be re-used' );
end
% Create marker array
[gYears, uYears] = findgroups(Years);
if max(gYears) > numel(markers)
    warning( 'More years than possible markers, markers will be re-used' );
end
% handle index out of range
markers = markers(mod((1:max(gYears))-1,numel(markers))+1);
colors = colors(mod((1:max(gInputs))-1,size(colors,1))+1,:);
% Setup as above...
lineProps = {'markersize', 5, 'linestyle', 'none', 'LineWidth', 2};
figure(); hold on
for iYr = 1:max(gYears)
    for iIn = 1:max(gInputs)
        idx = (iYr == gYears) & (iIn == gInputs); 
        line(t1Response(idx), t2Response(idx), 20, 'color', colors(iIn,:), 'Marker', markers{iYr}, 'handlevisibility', 'off', lineProps{:} );
    end
end
% Spoof markers for the legend
for iYr = 1:max(gYears)
    line( NaN, NaN, 'color', 'k', 'Marker', markers{iYr}, 'displayname', num2str(uYears(iYr)), lineProps{:} );
end
for iIn = 1:max(gInputs)
    line( NaN, NaN, 'color', colors(iIn,:), 'Marker', 'o', 'markersize', 20, 'displayname', num2str(uInputs(iIn)), lineProps{:} );
end
hold off
legend('show')